f(x)=a(x+-2)(x+-1(4+-4i))(x+-1(4+4i))

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Solution for f(x)=a(x+-2)(x+-1(4+-4i))(x+-1(4+4i)) equation: 


Simplifying
f(x) = a(x + -2)(x + -1(4 + -4i))(x + -1(4 + 4i))

Multiply f * x
fx = a(x + -2)(x + -1(4 + -4i))(x + -1(4 + 4i))

Reorder the terms:
fx = a(-2 + x)(x + -1(4 + -4i))(x + -1(4 + 4i))
fx = a(-2 + x)(x + (4 * -1 + -4i * -1))(x + -1(4 + 4i))
fx = a(-2 + x)(x + (-4 + 4i))(x + -1(4 + 4i))

Reorder the terms:
fx = a(-2 + x)(-4 + 4i + x)(x + -1(4 + 4i))
fx = a(-2 + x)(-4 + 4i + x)(x + (4 * -1 + 4i * -1))
fx = a(-2 + x)(-4 + 4i + x)(x + (-4 + -4i))

Reorder the terms:
fx = a(-2 + x)(-4 + 4i + x)(-4 + -4i + x)

Multiply (-2 + x) * (-4 + 4i + x)
fx = a(-2(-4 + 4i + x) + x(-4 + 4i + x))(-4 + -4i + x)
fx = a((-4 * -2 + 4i * -2 + x * -2) + x(-4 + 4i + x))(-4 + -4i + x)
fx = a((8 + -8i + -2x) + x(-4 + 4i + x))(-4 + -4i + x)
fx = a(8 + -8i + -2x + (-4 * x + 4i * x + x * x))(-4 + -4i + x)

Reorder the terms:
fx = a(8 + -8i + -2x + (4ix + -4x + x2))(-4 + -4i + x)
fx = a(8 + -8i + -2x + (4ix + -4x + x2))(-4 + -4i + x)

Reorder the terms:
fx = a(8 + -8i + 4ix + -2x + -4x + x2)(-4 + -4i + x)

Combine like terms: -2x + -4x = -6x
fx = a(8 + -8i + 4ix + -6x + x2)(-4 + -4i + x)

Multiply (8 + -8i + 4ix + -6x + x2) * (-4 + -4i + x)
fx = a(8(-4 + -4i + x) + -8i * (-4 + -4i + x) + 4ix * (-4 + -4i + x) + -6x * (-4 + -4i + x) + x2(-4 + -4i + x))
fx = a((-4 * 8 + -4i * 8 + x * 8) + -8i * (-4 + -4i + x) + 4ix * (-4 + -4i + x) + -6x * (-4 + -4i + x) + x2(-4 + -4i + x))
fx = a((-32 + -32i + 8x) + -8i * (-4 + -4i + x) + 4ix * (-4 + -4i + x) + -6x * (-4 + -4i + x) + x2(-4 + -4i + x))
fx = a(-32 + -32i + 8x + (-4 * -8i + -4i * -8i + x * -8i) + 4ix * (-4 + -4i + x) + -6x * (-4 + -4i + x) + x2(-4 + -4i + x))

Reorder the terms:
fx = a(-32 + -32i + 8x + (32i + -8ix + 32i2) + 4ix * (-4 + -4i + x) + -6x * (-4 + -4i + x) + x2(-4 + -4i + x))
fx = a(-32 + -32i + 8x + (32i + -8ix + 32i2) + 4ix * (-4 + -4i + x) + -6x * (-4 + -4i + x) + x2(-4 + -4i + x))
fx = a(-32 + -32i + 8x + 32i + -8ix + 32i2 + (-4 * 4ix + -4i * 4ix + x * 4ix) + -6x * (-4 + -4i + x) + x2(-4 + -4i + x))

Reorder the terms:
fx = a(-32 + -32i + 8x + 32i + -8ix + 32i2 + (-16ix + 4ix2 + -16i2x) + -6x * (-4 + -4i + x) + x2(-4 + -4i + x))
fx = a(-32 + -32i + 8x + 32i + -8ix + 32i2 + (-16ix + 4ix2 + -16i2x) + -6x * (-4 + -4i + x) + x2(-4 + -4i + x))
fx = a(-32 + -32i + 8x + 32i + -8ix + 32i2 + -16ix + 4ix2 + -16i2x + (-4 * -6x + -4i * -6x + x * -6x) + x2(-4 + -4i + x))

Reorder the terms:
fx = a(-32 + -32i + 8x + 32i + -8ix + 32i2 + -16ix + 4ix2 + -16i2x + (24ix + 24x + -6x2) + x2(-4 + -4i + x))
fx = a(-32 + -32i + 8x + 32i + -8ix + 32i2 + -16ix + 4ix2 + -16i2x + (24ix + 24x + -6x2) + x2(-4 + -4i + x))
fx = a(-32 + -32i + 8x + 32i + -8ix + 32i2 + -16ix + 4ix2 + -16i2x + 24ix + 24x + -6x2 + (-4 * x2 + -4i * x2 + x * x2))

Reorder the terms:
fx = a(-32 + -32i + 8x + 32i + -8ix + 32i2 + -16ix + 4ix2 + -16i2x + 24ix + 24x + -6x2 + (-4ix2 + -4x2 + x3))
fx = a(-32 + -32i + 8x + 32i + -8ix + 32i2 + -16ix + 4ix2 + -16i2x + 24ix + 24x + -6x2 + (-4ix2 + -4x2 + x3))

Reorder the terms:
fx = a(-32 + -32i + 32i + -8ix + -16ix + 24ix + 4ix2 + -4ix2 + 32i2 + -16i2x + 8x + 24x + -6x2 + -4x2 + x3)

Combine like terms: -32i + 32i = 0
fx = a(-32 + 0 + -8ix + -16ix + 24ix + 4ix2 + -4ix2 + 32i2 + -16i2x + 8x + 24x + -6x2 + -4x2 + x3)
fx = a(-32 + -8ix + -16ix + 24ix + 4ix2 + -4ix2 + 32i2 + -16i2x + 8x + 24x + -6x2 + -4x2 + x3)

Combine like terms: -8ix + -16ix = -24ix
fx = a(-32 + -24ix + 24ix + 4ix2 + -4ix2 + 32i2 + -16i2x + 8x + 24x + -6x2 + -4x2 + x3)

Combine like terms: -24ix + 24ix = 0
fx = a(-32 + 0 + 4ix2 + -4ix2 + 32i2 + -16i2x + 8x + 24x + -6x2 + -4x2 + x3)
fx = a(-32 + 4ix2 + -4ix2 + 32i2 + -16i2x + 8x + 24x + -6x2 + -4x2 + x3)

Combine like terms: 4ix2 + -4ix2 = 0
fx = a(-32 + 0 + 32i2 + -16i2x + 8x + 24x + -6x2 + -4x2 + x3)
fx = a(-32 + 32i2 + -16i2x + 8x + 24x + -6x2 + -4x2 + x3)

Combine like terms: 8x + 24x = 32x
fx = a(-32 + 32i2 + -16i2x + 32x + -6x2 + -4x2 + x3)

Combine like terms: -6x2 + -4x2 = -10x2
fx = a(-32 + 32i2 + -16i2x + 32x + -10x2 + x3)
fx = (-32 * a + 32i2 * a + -16i2x * a + 32x * a + -10x2 * a + x3 * a)
fx = (-32a + 32ai2 + -16ai2x + 32ax + -10ax2 + ax3)

Solving
fx = -32a + 32ai2 + -16ai2x + 32ax + -10ax2 + ax3

Solving for variable 'f'.

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

Divide each side by 'x'.
f = -32ax-1 + 32ai2x-1 + -16ai2 + 32a + -10ax + ax2

Simplifying
f = -32ax-1 + 32ai2x-1 + -16ai2 + 32a + -10ax + ax2

Reorder the terms:
f = 32a + -16ai2 + 32ai2x-1 + -32ax-1 + -10ax + ax2

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