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

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


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

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

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

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

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

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

Reorder the terms:
fx = a(1 + 5i + -1x + (-5i + 5ix + -25i2) + x(-1 + -5i + x))(4 + x)
fx = a(1 + 5i + -1x + (-5i + 5ix + -25i2) + x(-1 + -5i + x))(4 + x)
fx = a(1 + 5i + -1x + -5i + 5ix + -25i2 + (-1 * x + -5i * x + x * x))(4 + x)

Reorder the terms:
fx = a(1 + 5i + -1x + -5i + 5ix + -25i2 + (-5ix + -1x + x2))(4 + x)
fx = a(1 + 5i + -1x + -5i + 5ix + -25i2 + (-5ix + -1x + x2))(4 + x)

Reorder the terms:
fx = a(1 + 5i + -5i + 5ix + -5ix + -25i2 + -1x + -1x + x2)(4 + x)

Combine like terms: 5i + -5i = 0
fx = a(1 + 0 + 5ix + -5ix + -25i2 + -1x + -1x + x2)(4 + x)
fx = a(1 + 5ix + -5ix + -25i2 + -1x + -1x + x2)(4 + x)

Combine like terms: 5ix + -5ix = 0
fx = a(1 + 0 + -25i2 + -1x + -1x + x2)(4 + x)
fx = a(1 + -25i2 + -1x + -1x + x2)(4 + x)

Combine like terms: -1x + -1x = -2x
fx = a(1 + -25i2 + -2x + x2)(4 + x)

Multiply (1 + -25i2 + -2x + x2) * (4 + x)
fx = a(1(4 + x) + -25i2 * (4 + x) + -2x * (4 + x) + x2(4 + x))
fx = a((4 * 1 + x * 1) + -25i2 * (4 + x) + -2x * (4 + x) + x2(4 + x))
fx = a((4 + 1x) + -25i2 * (4 + x) + -2x * (4 + x) + x2(4 + x))
fx = a(4 + 1x + (4 * -25i2 + x * -25i2) + -2x * (4 + x) + x2(4 + x))
fx = a(4 + 1x + (-100i2 + -25i2x) + -2x * (4 + x) + x2(4 + x))
fx = a(4 + 1x + -100i2 + -25i2x + (4 * -2x + x * -2x) + x2(4 + x))
fx = a(4 + 1x + -100i2 + -25i2x + (-8x + -2x2) + x2(4 + x))
fx = a(4 + 1x + -100i2 + -25i2x + -8x + -2x2 + (4 * x2 + x * x2))
fx = a(4 + 1x + -100i2 + -25i2x + -8x + -2x2 + (4x2 + x3))

Reorder the terms:
fx = a(4 + -100i2 + -25i2x + 1x + -8x + -2x2 + 4x2 + x3)

Combine like terms: 1x + -8x = -7x
fx = a(4 + -100i2 + -25i2x + -7x + -2x2 + 4x2 + x3)

Combine like terms: -2x2 + 4x2 = 2x2
fx = a(4 + -100i2 + -25i2x + -7x + 2x2 + x3)
fx = (4 * a + -100i2 * a + -25i2x * a + -7x * a + 2x2 * a + x3 * a)
fx = (4a + -100ai2 + -25ai2x + -7ax + 2ax2 + ax3)

Solving
fx = 4a + -100ai2 + -25ai2x + -7ax + 2ax2 + 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 = 4ax-1 + -100ai2x-1 + -25ai2 + -7a + 2ax + ax2

Simplifying
f = 4ax-1 + -100ai2x-1 + -25ai2 + -7a + 2ax + ax2

Reorder the terms:
f = -7a + -25ai2 + -100ai2x-1 + 4ax-1 + 2ax + ax2

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