f(x)=a*(x-5)(x-5)(x-(4-4i))*(x-(4+4i))

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Solution for f(x)=a*(x-5)(x-5)(x-(4-4i))*(x-(4+4i)) equation: 


Simplifying
f(x) = a(x + -5)(x + -5)(x + -1(4 + -4i))(x + -1(4 + 4i))

Multiply f * x
fx = a(x + -5)(x + -5)(x + -1(4 + -4i))(x + -1(4 + 4i))

Reorder the terms:
fx = a(-5 + x)(x + -5)(x + -1(4 + -4i))(x + -1(4 + 4i))

Reorder the terms:
fx = a(-5 + x)(-5 + x)(x + -1(4 + -4i))(x + -1(4 + 4i))
fx = a(-5 + x)(-5 + x)(x + (4 * -1 + -4i * -1))(x + -1(4 + 4i))
fx = a(-5 + x)(-5 + x)(x + (-4 + 4i))(x + -1(4 + 4i))

Reorder the terms:
fx = a(-5 + x)(-5 + x)(-4 + 4i + x)(x + -1(4 + 4i))
fx = a(-5 + x)(-5 + x)(-4 + 4i + x)(x + (4 * -1 + 4i * -1))
fx = a(-5 + x)(-5 + x)(-4 + 4i + x)(x + (-4 + -4i))

Reorder the terms:
fx = a(-5 + x)(-5 + x)(-4 + 4i + x)(-4 + -4i + x)

Multiply (-5 + x) * (-5 + x)
fx = a(-5(-5 + x) + x(-5 + x))(-4 + 4i + x)(-4 + -4i + x)
fx = a((-5 * -5 + x * -5) + x(-5 + x))(-4 + 4i + x)(-4 + -4i + x)
fx = a((25 + -5x) + x(-5 + x))(-4 + 4i + x)(-4 + -4i + x)
fx = a(25 + -5x + (-5 * x + x * x))(-4 + 4i + x)(-4 + -4i + x)
fx = a(25 + -5x + (-5x + x2))(-4 + 4i + x)(-4 + -4i + x)

Combine like terms: -5x + -5x = -10x
fx = a(25 + -10x + x2)(-4 + 4i + x)(-4 + -4i + x)

Multiply (25 + -10x + x2) * (-4 + 4i + x)
fx = a(25(-4 + 4i + x) + -10x * (-4 + 4i + x) + x2(-4 + 4i + x))(-4 + -4i + x)
fx = a((-4 * 25 + 4i * 25 + x * 25) + -10x * (-4 + 4i + x) + x2(-4 + 4i + x))(-4 + -4i + x)
fx = a((-100 + 100i + 25x) + -10x * (-4 + 4i + x) + x2(-4 + 4i + x))(-4 + -4i + x)
fx = a(-100 + 100i + 25x + (-4 * -10x + 4i * -10x + x * -10x) + x2(-4 + 4i + x))(-4 + -4i + x)

Reorder the terms:
fx = a(-100 + 100i + 25x + (-40ix + 40x + -10x2) + x2(-4 + 4i + x))(-4 + -4i + x)
fx = a(-100 + 100i + 25x + (-40ix + 40x + -10x2) + x2(-4 + 4i + x))(-4 + -4i + x)
fx = a(-100 + 100i + 25x + -40ix + 40x + -10x2 + (-4 * x2 + 4i * x2 + x * x2))(-4 + -4i + x)

Reorder the terms:
fx = a(-100 + 100i + 25x + -40ix + 40x + -10x2 + (4ix2 + -4x2 + x3))(-4 + -4i + x)
fx = a(-100 + 100i + 25x + -40ix + 40x + -10x2 + (4ix2 + -4x2 + x3))(-4 + -4i + x)

Reorder the terms:
fx = a(-100 + 100i + -40ix + 4ix2 + 25x + 40x + -10x2 + -4x2 + x3)(-4 + -4i + x)

Combine like terms: 25x + 40x = 65x
fx = a(-100 + 100i + -40ix + 4ix2 + 65x + -10x2 + -4x2 + x3)(-4 + -4i + x)

Combine like terms: -10x2 + -4x2 = -14x2
fx = a(-100 + 100i + -40ix + 4ix2 + 65x + -14x2 + x3)(-4 + -4i + x)

Multiply (-100 + 100i + -40ix + 4ix2 + 65x + -14x2 + x3) * (-4 + -4i + x)
fx = a(-100(-4 + -4i + x) + 100i * (-4 + -4i + x) + -40ix * (-4 + -4i + x) + 4ix2 * (-4 + -4i + x) + 65x * (-4 + -4i + x) + -14x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a((-4 * -100 + -4i * -100 + x * -100) + 100i * (-4 + -4i + x) + -40ix * (-4 + -4i + x) + 4ix2 * (-4 + -4i + x) + 65x * (-4 + -4i + x) + -14x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a((400 + 400i + -100x) + 100i * (-4 + -4i + x) + -40ix * (-4 + -4i + x) + 4ix2 * (-4 + -4i + x) + 65x * (-4 + -4i + x) + -14x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(400 + 400i + -100x + (-4 * 100i + -4i * 100i + x * 100i) + -40ix * (-4 + -4i + x) + 4ix2 * (-4 + -4i + x) + 65x * (-4 + -4i + x) + -14x2 * (-4 + -4i + x) + x3(-4 + -4i + x))

Reorder the terms:
fx = a(400 + 400i + -100x + (-400i + 100ix + -400i2) + -40ix * (-4 + -4i + x) + 4ix2 * (-4 + -4i + x) + 65x * (-4 + -4i + x) + -14x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(400 + 400i + -100x + (-400i + 100ix + -400i2) + -40ix * (-4 + -4i + x) + 4ix2 * (-4 + -4i + x) + 65x * (-4 + -4i + x) + -14x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(400 + 400i + -100x + -400i + 100ix + -400i2 + (-4 * -40ix + -4i * -40ix + x * -40ix) + 4ix2 * (-4 + -4i + x) + 65x * (-4 + -4i + x) + -14x2 * (-4 + -4i + x) + x3(-4 + -4i + x))

Reorder the terms:
fx = a(400 + 400i + -100x + -400i + 100ix + -400i2 + (160ix + -40ix2 + 160i2x) + 4ix2 * (-4 + -4i + x) + 65x * (-4 + -4i + x) + -14x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(400 + 400i + -100x + -400i + 100ix + -400i2 + (160ix + -40ix2 + 160i2x) + 4ix2 * (-4 + -4i + x) + 65x * (-4 + -4i + x) + -14x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(400 + 400i + -100x + -400i + 100ix + -400i2 + 160ix + -40ix2 + 160i2x + (-4 * 4ix2 + -4i * 4ix2 + x * 4ix2) + 65x * (-4 + -4i + x) + -14x2 * (-4 + -4i + x) + x3(-4 + -4i + x))

Reorder the terms:
fx = a(400 + 400i + -100x + -400i + 100ix + -400i2 + 160ix + -40ix2 + 160i2x + (-16ix2 + 4ix3 + -16i2x2) + 65x * (-4 + -4i + x) + -14x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(400 + 400i + -100x + -400i + 100ix + -400i2 + 160ix + -40ix2 + 160i2x + (-16ix2 + 4ix3 + -16i2x2) + 65x * (-4 + -4i + x) + -14x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(400 + 400i + -100x + -400i + 100ix + -400i2 + 160ix + -40ix2 + 160i2x + -16ix2 + 4ix3 + -16i2x2 + (-4 * 65x + -4i * 65x + x * 65x) + -14x2 * (-4 + -4i + x) + x3(-4 + -4i + x))

Reorder the terms:
fx = a(400 + 400i + -100x + -400i + 100ix + -400i2 + 160ix + -40ix2 + 160i2x + -16ix2 + 4ix3 + -16i2x2 + (-260ix + -260x + 65x2) + -14x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(400 + 400i + -100x + -400i + 100ix + -400i2 + 160ix + -40ix2 + 160i2x + -16ix2 + 4ix3 + -16i2x2 + (-260ix + -260x + 65x2) + -14x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(400 + 400i + -100x + -400i + 100ix + -400i2 + 160ix + -40ix2 + 160i2x + -16ix2 + 4ix3 + -16i2x2 + -260ix + -260x + 65x2 + (-4 * -14x2 + -4i * -14x2 + x * -14x2) + x3(-4 + -4i + x))

Reorder the terms:
fx = a(400 + 400i + -100x + -400i + 100ix + -400i2 + 160ix + -40ix2 + 160i2x + -16ix2 + 4ix3 + -16i2x2 + -260ix + -260x + 65x2 + (56ix2 + 56x2 + -14x3) + x3(-4 + -4i + x))
fx = a(400 + 400i + -100x + -400i + 100ix + -400i2 + 160ix + -40ix2 + 160i2x + -16ix2 + 4ix3 + -16i2x2 + -260ix + -260x + 65x2 + (56ix2 + 56x2 + -14x3) + x3(-4 + -4i + x))
fx = a(400 + 400i + -100x + -400i + 100ix + -400i2 + 160ix + -40ix2 + 160i2x + -16ix2 + 4ix3 + -16i2x2 + -260ix + -260x + 65x2 + 56ix2 + 56x2 + -14x3 + (-4 * x3 + -4i * x3 + x * x3))

Reorder the terms:
fx = a(400 + 400i + -100x + -400i + 100ix + -400i2 + 160ix + -40ix2 + 160i2x + -16ix2 + 4ix3 + -16i2x2 + -260ix + -260x + 65x2 + 56ix2 + 56x2 + -14x3 + (-4ix3 + -4x3 + x4))
fx = a(400 + 400i + -100x + -400i + 100ix + -400i2 + 160ix + -40ix2 + 160i2x + -16ix2 + 4ix3 + -16i2x2 + -260ix + -260x + 65x2 + 56ix2 + 56x2 + -14x3 + (-4ix3 + -4x3 + x4))

Reorder the terms:
fx = a(400 + 400i + -400i + 100ix + 160ix + -260ix + -40ix2 + -16ix2 + 56ix2 + 4ix3 + -4ix3 + -400i2 + 160i2x + -16i2x2 + -100x + -260x + 65x2 + 56x2 + -14x3 + -4x3 + x4)

Combine like terms: 400i + -400i = 0
fx = a(400 + 0 + 100ix + 160ix + -260ix + -40ix2 + -16ix2 + 56ix2 + 4ix3 + -4ix3 + -400i2 + 160i2x + -16i2x2 + -100x + -260x + 65x2 + 56x2 + -14x3 + -4x3 + x4)
fx = a(400 + 100ix + 160ix + -260ix + -40ix2 + -16ix2 + 56ix2 + 4ix3 + -4ix3 + -400i2 + 160i2x + -16i2x2 + -100x + -260x + 65x2 + 56x2 + -14x3 + -4x3 + x4)

Combine like terms: 100ix + 160ix = 260ix
fx = a(400 + 260ix + -260ix + -40ix2 + -16ix2 + 56ix2 + 4ix3 + -4ix3 + -400i2 + 160i2x + -16i2x2 + -100x + -260x + 65x2 + 56x2 + -14x3 + -4x3 + x4)

Combine like terms: 260ix + -260ix = 0
fx = a(400 + 0 + -40ix2 + -16ix2 + 56ix2 + 4ix3 + -4ix3 + -400i2 + 160i2x + -16i2x2 + -100x + -260x + 65x2 + 56x2 + -14x3 + -4x3 + x4)
fx = a(400 + -40ix2 + -16ix2 + 56ix2 + 4ix3 + -4ix3 + -400i2 + 160i2x + -16i2x2 + -100x + -260x + 65x2 + 56x2 + -14x3 + -4x3 + x4)

Combine like terms: -40ix2 + -16ix2 = -56ix2
fx = a(400 + -56ix2 + 56ix2 + 4ix3 + -4ix3 + -400i2 + 160i2x + -16i2x2 + -100x + -260x + 65x2 + 56x2 + -14x3 + -4x3 + x4)

Combine like terms: -56ix2 + 56ix2 = 0
fx = a(400 + 0 + 4ix3 + -4ix3 + -400i2 + 160i2x + -16i2x2 + -100x + -260x + 65x2 + 56x2 + -14x3 + -4x3 + x4)
fx = a(400 + 4ix3 + -4ix3 + -400i2 + 160i2x + -16i2x2 + -100x + -260x + 65x2 + 56x2 + -14x3 + -4x3 + x4)

Combine like terms: 4ix3 + -4ix3 = 0
fx = a(400 + 0 + -400i2 + 160i2x + -16i2x2 + -100x + -260x + 65x2 + 56x2 + -14x3 + -4x3 + x4)
fx = a(400 + -400i2 + 160i2x + -16i2x2 + -100x + -260x + 65x2 + 56x2 + -14x3 + -4x3 + x4)

Combine like terms: -100x + -260x = -360x
fx = a(400 + -400i2 + 160i2x + -16i2x2 + -360x + 65x2 + 56x2 + -14x3 + -4x3 + x4)

Combine like terms: 65x2 + 56x2 = 121x2
fx = a(400 + -400i2 + 160i2x + -16i2x2 + -360x + 121x2 + -14x3 + -4x3 + x4)

Combine like terms: -14x3 + -4x3 = -18x3
fx = a(400 + -400i2 + 160i2x + -16i2x2 + -360x + 121x2 + -18x3 + x4)
fx = (400 * a + -400i2 * a + 160i2x * a + -16i2x2 * a + -360x * a + 121x2 * a + -18x3 * a + x4 * a)
fx = (400a + -400ai2 + 160ai2x + -16ai2x2 + -360ax + 121ax2 + -18ax3 + ax4)

Solving
fx = 400a + -400ai2 + 160ai2x + -16ai2x2 + -360ax + 121ax2 + -18ax3 + ax4

Solving for variable 'f'.

Move all terms containing f to the left, all other terms to the right.

Divide each side by 'x'.
f = 400ax-1 + -400ai2x-1 + 160ai2 + -16ai2x + -360a + 121ax + -18ax2 + ax3

Simplifying
f = 400ax-1 + -400ai2x-1 + 160ai2 + -16ai2x + -360a + 121ax + -18ax2 + ax3

Reorder the terms:
f = -360a + 160ai2 + -400ai2x-1 + -16ai2x + 400ax-1 + 121ax + -18ax2 + ax3

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