f(x)=a(x+2)(x-5+4i)(x-5-4i)

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Solution for f(x)=a(x+2)(x-5+4i)(x-5-4i) equation: 


Simplifying
f(x) = a(x + 2)(x + -5 + 4i)(x + -5 + -4i)

Multiply f * x
fx = a(x + 2)(x + -5 + 4i)(x + -5 + -4i)

Reorder the terms:
fx = a(2 + x)(x + -5 + 4i)(x + -5 + -4i)

Reorder the terms:
fx = a(2 + x)(-5 + 4i + x)(x + -5 + -4i)

Reorder the terms:
fx = a(2 + x)(-5 + 4i + x)(-5 + -4i + x)

Multiply (2 + x) * (-5 + 4i + x)
fx = a(2(-5 + 4i + x) + x(-5 + 4i + x))(-5 + -4i + x)
fx = a((-5 * 2 + 4i * 2 + x * 2) + x(-5 + 4i + x))(-5 + -4i + x)
fx = a((-10 + 8i + 2x) + x(-5 + 4i + x))(-5 + -4i + x)
fx = a(-10 + 8i + 2x + (-5 * x + 4i * x + x * x))(-5 + -4i + x)

Reorder the terms:
fx = a(-10 + 8i + 2x + (4ix + -5x + x2))(-5 + -4i + x)
fx = a(-10 + 8i + 2x + (4ix + -5x + x2))(-5 + -4i + x)

Reorder the terms:
fx = a(-10 + 8i + 4ix + 2x + -5x + x2)(-5 + -4i + x)

Combine like terms: 2x + -5x = -3x
fx = a(-10 + 8i + 4ix + -3x + x2)(-5 + -4i + x)

Multiply (-10 + 8i + 4ix + -3x + x2) * (-5 + -4i + x)
fx = a(-10(-5 + -4i + x) + 8i * (-5 + -4i + x) + 4ix * (-5 + -4i + x) + -3x * (-5 + -4i + x) + x2(-5 + -4i + x))
fx = a((-5 * -10 + -4i * -10 + x * -10) + 8i * (-5 + -4i + x) + 4ix * (-5 + -4i + x) + -3x * (-5 + -4i + x) + x2(-5 + -4i + x))
fx = a((50 + 40i + -10x) + 8i * (-5 + -4i + x) + 4ix * (-5 + -4i + x) + -3x * (-5 + -4i + x) + x2(-5 + -4i + x))
fx = a(50 + 40i + -10x + (-5 * 8i + -4i * 8i + x * 8i) + 4ix * (-5 + -4i + x) + -3x * (-5 + -4i + x) + x2(-5 + -4i + x))

Reorder the terms:
fx = a(50 + 40i + -10x + (-40i + 8ix + -32i2) + 4ix * (-5 + -4i + x) + -3x * (-5 + -4i + x) + x2(-5 + -4i + x))
fx = a(50 + 40i + -10x + (-40i + 8ix + -32i2) + 4ix * (-5 + -4i + x) + -3x * (-5 + -4i + x) + x2(-5 + -4i + x))
fx = a(50 + 40i + -10x + -40i + 8ix + -32i2 + (-5 * 4ix + -4i * 4ix + x * 4ix) + -3x * (-5 + -4i + x) + x2(-5 + -4i + x))

Reorder the terms:
fx = a(50 + 40i + -10x + -40i + 8ix + -32i2 + (-20ix + 4ix2 + -16i2x) + -3x * (-5 + -4i + x) + x2(-5 + -4i + x))
fx = a(50 + 40i + -10x + -40i + 8ix + -32i2 + (-20ix + 4ix2 + -16i2x) + -3x * (-5 + -4i + x) + x2(-5 + -4i + x))
fx = a(50 + 40i + -10x + -40i + 8ix + -32i2 + -20ix + 4ix2 + -16i2x + (-5 * -3x + -4i * -3x + x * -3x) + x2(-5 + -4i + x))

Reorder the terms:
fx = a(50 + 40i + -10x + -40i + 8ix + -32i2 + -20ix + 4ix2 + -16i2x + (12ix + 15x + -3x2) + x2(-5 + -4i + x))
fx = a(50 + 40i + -10x + -40i + 8ix + -32i2 + -20ix + 4ix2 + -16i2x + (12ix + 15x + -3x2) + x2(-5 + -4i + x))
fx = a(50 + 40i + -10x + -40i + 8ix + -32i2 + -20ix + 4ix2 + -16i2x + 12ix + 15x + -3x2 + (-5 * x2 + -4i * x2 + x * x2))

Reorder the terms:
fx = a(50 + 40i + -10x + -40i + 8ix + -32i2 + -20ix + 4ix2 + -16i2x + 12ix + 15x + -3x2 + (-4ix2 + -5x2 + x3))
fx = a(50 + 40i + -10x + -40i + 8ix + -32i2 + -20ix + 4ix2 + -16i2x + 12ix + 15x + -3x2 + (-4ix2 + -5x2 + x3))

Reorder the terms:
fx = a(50 + 40i + -40i + 8ix + -20ix + 12ix + 4ix2 + -4ix2 + -32i2 + -16i2x + -10x + 15x + -3x2 + -5x2 + x3)

Combine like terms: 40i + -40i = 0
fx = a(50 + 0 + 8ix + -20ix + 12ix + 4ix2 + -4ix2 + -32i2 + -16i2x + -10x + 15x + -3x2 + -5x2 + x3)
fx = a(50 + 8ix + -20ix + 12ix + 4ix2 + -4ix2 + -32i2 + -16i2x + -10x + 15x + -3x2 + -5x2 + x3)

Combine like terms: 8ix + -20ix = -12ix
fx = a(50 + -12ix + 12ix + 4ix2 + -4ix2 + -32i2 + -16i2x + -10x + 15x + -3x2 + -5x2 + x3)

Combine like terms: -12ix + 12ix = 0
fx = a(50 + 0 + 4ix2 + -4ix2 + -32i2 + -16i2x + -10x + 15x + -3x2 + -5x2 + x3)
fx = a(50 + 4ix2 + -4ix2 + -32i2 + -16i2x + -10x + 15x + -3x2 + -5x2 + x3)

Combine like terms: 4ix2 + -4ix2 = 0
fx = a(50 + 0 + -32i2 + -16i2x + -10x + 15x + -3x2 + -5x2 + x3)
fx = a(50 + -32i2 + -16i2x + -10x + 15x + -3x2 + -5x2 + x3)

Combine like terms: -10x + 15x = 5x
fx = a(50 + -32i2 + -16i2x + 5x + -3x2 + -5x2 + x3)

Combine like terms: -3x2 + -5x2 = -8x2
fx = a(50 + -32i2 + -16i2x + 5x + -8x2 + x3)
fx = (50 * a + -32i2 * a + -16i2x * a + 5x * a + -8x2 * a + x3 * a)
fx = (50a + -32ai2 + -16ai2x + 5ax + -8ax2 + ax3)

Solving
fx = 50a + -32ai2 + -16ai2x + 5ax + -8ax2 + 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 = 50ax-1 + -32ai2x-1 + -16ai2 + 5a + -8ax + ax2

Simplifying
f = 50ax-1 + -32ai2x-1 + -16ai2 + 5a + -8ax + ax2

Reorder the terms:
f = 5a + -16ai2 + -32ai2x-1 + 50ax-1 + -8ax + ax2

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