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

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Solution for f(x)=a(x+1)(x-5)(x+-1(4+-4i))(x+-1(4+4i)) equation: 


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

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

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

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

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

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

Multiply (1 + x) * (-5 + x)
fx = a(1(-5 + x) + x(-5 + x))(-4 + 4i + x)(-4 + -4i + x)
fx = a((-5 * 1 + x * 1) + x(-5 + x))(-4 + 4i + x)(-4 + -4i + x)
fx = a((-5 + 1x) + x(-5 + x))(-4 + 4i + x)(-4 + -4i + x)
fx = a(-5 + 1x + (-5 * x + x * x))(-4 + 4i + x)(-4 + -4i + x)
fx = a(-5 + 1x + (-5x + x2))(-4 + 4i + x)(-4 + -4i + x)

Combine like terms: 1x + -5x = -4x
fx = a(-5 + -4x + x2)(-4 + 4i + x)(-4 + -4i + x)

Multiply (-5 + -4x + x2) * (-4 + 4i + x)
fx = a(-5(-4 + 4i + x) + -4x * (-4 + 4i + x) + x2(-4 + 4i + x))(-4 + -4i + x)
fx = a((-4 * -5 + 4i * -5 + x * -5) + -4x * (-4 + 4i + x) + x2(-4 + 4i + x))(-4 + -4i + x)
fx = a((20 + -20i + -5x) + -4x * (-4 + 4i + x) + x2(-4 + 4i + x))(-4 + -4i + x)
fx = a(20 + -20i + -5x + (-4 * -4x + 4i * -4x + x * -4x) + x2(-4 + 4i + x))(-4 + -4i + x)

Reorder the terms:
fx = a(20 + -20i + -5x + (-16ix + 16x + -4x2) + x2(-4 + 4i + x))(-4 + -4i + x)
fx = a(20 + -20i + -5x + (-16ix + 16x + -4x2) + x2(-4 + 4i + x))(-4 + -4i + x)
fx = a(20 + -20i + -5x + -16ix + 16x + -4x2 + (-4 * x2 + 4i * x2 + x * x2))(-4 + -4i + x)

Reorder the terms:
fx = a(20 + -20i + -5x + -16ix + 16x + -4x2 + (4ix2 + -4x2 + x3))(-4 + -4i + x)
fx = a(20 + -20i + -5x + -16ix + 16x + -4x2 + (4ix2 + -4x2 + x3))(-4 + -4i + x)

Reorder the terms:
fx = a(20 + -20i + -16ix + 4ix2 + -5x + 16x + -4x2 + -4x2 + x3)(-4 + -4i + x)

Combine like terms: -5x + 16x = 11x
fx = a(20 + -20i + -16ix + 4ix2 + 11x + -4x2 + -4x2 + x3)(-4 + -4i + x)

Combine like terms: -4x2 + -4x2 = -8x2
fx = a(20 + -20i + -16ix + 4ix2 + 11x + -8x2 + x3)(-4 + -4i + x)

Multiply (20 + -20i + -16ix + 4ix2 + 11x + -8x2 + x3) * (-4 + -4i + x)
fx = a(20(-4 + -4i + x) + -20i * (-4 + -4i + x) + -16ix * (-4 + -4i + x) + 4ix2 * (-4 + -4i + x) + 11x * (-4 + -4i + x) + -8x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a((-4 * 20 + -4i * 20 + x * 20) + -20i * (-4 + -4i + x) + -16ix * (-4 + -4i + x) + 4ix2 * (-4 + -4i + x) + 11x * (-4 + -4i + x) + -8x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a((-80 + -80i + 20x) + -20i * (-4 + -4i + x) + -16ix * (-4 + -4i + x) + 4ix2 * (-4 + -4i + x) + 11x * (-4 + -4i + x) + -8x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(-80 + -80i + 20x + (-4 * -20i + -4i * -20i + x * -20i) + -16ix * (-4 + -4i + x) + 4ix2 * (-4 + -4i + x) + 11x * (-4 + -4i + x) + -8x2 * (-4 + -4i + x) + x3(-4 + -4i + x))

Reorder the terms:
fx = a(-80 + -80i + 20x + (80i + -20ix + 80i2) + -16ix * (-4 + -4i + x) + 4ix2 * (-4 + -4i + x) + 11x * (-4 + -4i + x) + -8x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(-80 + -80i + 20x + (80i + -20ix + 80i2) + -16ix * (-4 + -4i + x) + 4ix2 * (-4 + -4i + x) + 11x * (-4 + -4i + x) + -8x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(-80 + -80i + 20x + 80i + -20ix + 80i2 + (-4 * -16ix + -4i * -16ix + x * -16ix) + 4ix2 * (-4 + -4i + x) + 11x * (-4 + -4i + x) + -8x2 * (-4 + -4i + x) + x3(-4 + -4i + x))

Reorder the terms:
fx = a(-80 + -80i + 20x + 80i + -20ix + 80i2 + (64ix + -16ix2 + 64i2x) + 4ix2 * (-4 + -4i + x) + 11x * (-4 + -4i + x) + -8x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(-80 + -80i + 20x + 80i + -20ix + 80i2 + (64ix + -16ix2 + 64i2x) + 4ix2 * (-4 + -4i + x) + 11x * (-4 + -4i + x) + -8x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(-80 + -80i + 20x + 80i + -20ix + 80i2 + 64ix + -16ix2 + 64i2x + (-4 * 4ix2 + -4i * 4ix2 + x * 4ix2) + 11x * (-4 + -4i + x) + -8x2 * (-4 + -4i + x) + x3(-4 + -4i + x))

Reorder the terms:
fx = a(-80 + -80i + 20x + 80i + -20ix + 80i2 + 64ix + -16ix2 + 64i2x + (-16ix2 + 4ix3 + -16i2x2) + 11x * (-4 + -4i + x) + -8x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(-80 + -80i + 20x + 80i + -20ix + 80i2 + 64ix + -16ix2 + 64i2x + (-16ix2 + 4ix3 + -16i2x2) + 11x * (-4 + -4i + x) + -8x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(-80 + -80i + 20x + 80i + -20ix + 80i2 + 64ix + -16ix2 + 64i2x + -16ix2 + 4ix3 + -16i2x2 + (-4 * 11x + -4i * 11x + x * 11x) + -8x2 * (-4 + -4i + x) + x3(-4 + -4i + x))

Reorder the terms:
fx = a(-80 + -80i + 20x + 80i + -20ix + 80i2 + 64ix + -16ix2 + 64i2x + -16ix2 + 4ix3 + -16i2x2 + (-44ix + -44x + 11x2) + -8x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(-80 + -80i + 20x + 80i + -20ix + 80i2 + 64ix + -16ix2 + 64i2x + -16ix2 + 4ix3 + -16i2x2 + (-44ix + -44x + 11x2) + -8x2 * (-4 + -4i + x) + x3(-4 + -4i + x))
fx = a(-80 + -80i + 20x + 80i + -20ix + 80i2 + 64ix + -16ix2 + 64i2x + -16ix2 + 4ix3 + -16i2x2 + -44ix + -44x + 11x2 + (-4 * -8x2 + -4i * -8x2 + x * -8x2) + x3(-4 + -4i + x))

Reorder the terms:
fx = a(-80 + -80i + 20x + 80i + -20ix + 80i2 + 64ix + -16ix2 + 64i2x + -16ix2 + 4ix3 + -16i2x2 + -44ix + -44x + 11x2 + (32ix2 + 32x2 + -8x3) + x3(-4 + -4i + x))
fx = a(-80 + -80i + 20x + 80i + -20ix + 80i2 + 64ix + -16ix2 + 64i2x + -16ix2 + 4ix3 + -16i2x2 + -44ix + -44x + 11x2 + (32ix2 + 32x2 + -8x3) + x3(-4 + -4i + x))
fx = a(-80 + -80i + 20x + 80i + -20ix + 80i2 + 64ix + -16ix2 + 64i2x + -16ix2 + 4ix3 + -16i2x2 + -44ix + -44x + 11x2 + 32ix2 + 32x2 + -8x3 + (-4 * x3 + -4i * x3 + x * x3))

Reorder the terms:
fx = a(-80 + -80i + 20x + 80i + -20ix + 80i2 + 64ix + -16ix2 + 64i2x + -16ix2 + 4ix3 + -16i2x2 + -44ix + -44x + 11x2 + 32ix2 + 32x2 + -8x3 + (-4ix3 + -4x3 + x4))
fx = a(-80 + -80i + 20x + 80i + -20ix + 80i2 + 64ix + -16ix2 + 64i2x + -16ix2 + 4ix3 + -16i2x2 + -44ix + -44x + 11x2 + 32ix2 + 32x2 + -8x3 + (-4ix3 + -4x3 + x4))

Reorder the terms:
fx = a(-80 + -80i + 80i + -20ix + 64ix + -44ix + -16ix2 + -16ix2 + 32ix2 + 4ix3 + -4ix3 + 80i2 + 64i2x + -16i2x2 + 20x + -44x + 11x2 + 32x2 + -8x3 + -4x3 + x4)

Combine like terms: -80i + 80i = 0
fx = a(-80 + 0 + -20ix + 64ix + -44ix + -16ix2 + -16ix2 + 32ix2 + 4ix3 + -4ix3 + 80i2 + 64i2x + -16i2x2 + 20x + -44x + 11x2 + 32x2 + -8x3 + -4x3 + x4)
fx = a(-80 + -20ix + 64ix + -44ix + -16ix2 + -16ix2 + 32ix2 + 4ix3 + -4ix3 + 80i2 + 64i2x + -16i2x2 + 20x + -44x + 11x2 + 32x2 + -8x3 + -4x3 + x4)

Combine like terms: -20ix + 64ix = 44ix
fx = a(-80 + 44ix + -44ix + -16ix2 + -16ix2 + 32ix2 + 4ix3 + -4ix3 + 80i2 + 64i2x + -16i2x2 + 20x + -44x + 11x2 + 32x2 + -8x3 + -4x3 + x4)

Combine like terms: 44ix + -44ix = 0
fx = a(-80 + 0 + -16ix2 + -16ix2 + 32ix2 + 4ix3 + -4ix3 + 80i2 + 64i2x + -16i2x2 + 20x + -44x + 11x2 + 32x2 + -8x3 + -4x3 + x4)
fx = a(-80 + -16ix2 + -16ix2 + 32ix2 + 4ix3 + -4ix3 + 80i2 + 64i2x + -16i2x2 + 20x + -44x + 11x2 + 32x2 + -8x3 + -4x3 + x4)

Combine like terms: -16ix2 + -16ix2 = -32ix2
fx = a(-80 + -32ix2 + 32ix2 + 4ix3 + -4ix3 + 80i2 + 64i2x + -16i2x2 + 20x + -44x + 11x2 + 32x2 + -8x3 + -4x3 + x4)

Combine like terms: -32ix2 + 32ix2 = 0
fx = a(-80 + 0 + 4ix3 + -4ix3 + 80i2 + 64i2x + -16i2x2 + 20x + -44x + 11x2 + 32x2 + -8x3 + -4x3 + x4)
fx = a(-80 + 4ix3 + -4ix3 + 80i2 + 64i2x + -16i2x2 + 20x + -44x + 11x2 + 32x2 + -8x3 + -4x3 + x4)

Combine like terms: 4ix3 + -4ix3 = 0
fx = a(-80 + 0 + 80i2 + 64i2x + -16i2x2 + 20x + -44x + 11x2 + 32x2 + -8x3 + -4x3 + x4)
fx = a(-80 + 80i2 + 64i2x + -16i2x2 + 20x + -44x + 11x2 + 32x2 + -8x3 + -4x3 + x4)

Combine like terms: 20x + -44x = -24x
fx = a(-80 + 80i2 + 64i2x + -16i2x2 + -24x + 11x2 + 32x2 + -8x3 + -4x3 + x4)

Combine like terms: 11x2 + 32x2 = 43x2
fx = a(-80 + 80i2 + 64i2x + -16i2x2 + -24x + 43x2 + -8x3 + -4x3 + x4)

Combine like terms: -8x3 + -4x3 = -12x3
fx = a(-80 + 80i2 + 64i2x + -16i2x2 + -24x + 43x2 + -12x3 + x4)
fx = (-80 * a + 80i2 * a + 64i2x * a + -16i2x2 * a + -24x * a + 43x2 * a + -12x3 * a + x4 * a)
fx = (-80a + 80ai2 + 64ai2x + -16ai2x2 + -24ax + 43ax2 + -12ax3 + ax4)

Solving
fx = -80a + 80ai2 + 64ai2x + -16ai2x2 + -24ax + 43ax2 + -12ax3 + 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 = -80ax-1 + 80ai2x-1 + 64ai2 + -16ai2x + -24a + 43ax + -12ax2 + ax3

Simplifying
f = -80ax-1 + 80ai2x-1 + 64ai2 + -16ai2x + -24a + 43ax + -12ax2 + ax3

Reorder the terms:
f = -24a + 64ai2 + 80ai2x-1 + -16ai2x + -80ax-1 + 43ax + -12ax2 + ax3

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