f(x)=(x-2)(x+3)(x-4i)(x+4i)

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Solution for f(x)=(x-2)(x+3)(x-4i)(x+4i) equation: 


Simplifying
f(x) = (x + -2)(x + 3)(x + -4i)(x + 4i)

Multiply f * x
fx = (x + -2)(x + 3)(x + -4i)(x + 4i)

Reorder the terms:
fx = (-2 + x)(x + 3)(x + -4i)(x + 4i)

Reorder the terms:
fx = (-2 + x)(3 + x)(x + -4i)(x + 4i)

Reorder the terms:
fx = (-2 + x)(3 + x)(-4i + x)(x + 4i)

Reorder the terms:
fx = (-2 + x)(3 + x)(-4i + x)(4i + x)

Multiply (-2 + x) * (3 + x)
fx = (-2(3 + x) + x(3 + x))(-4i + x)(4i + x)
fx = ((3 * -2 + x * -2) + x(3 + x))(-4i + x)(4i + x)
fx = ((-6 + -2x) + x(3 + x))(-4i + x)(4i + x)
fx = (-6 + -2x + (3 * x + x * x))(-4i + x)(4i + x)
fx = (-6 + -2x + (3x + x2))(-4i + x)(4i + x)

Combine like terms: -2x + 3x = 1x
fx = (-6 + 1x + x2)(-4i + x)(4i + x)

Multiply (-6 + 1x + x2) * (-4i + x)
fx = (-6(-4i + x) + 1x * (-4i + x) + x2(-4i + x))(4i + x)
fx = ((-4i * -6 + x * -6) + 1x * (-4i + x) + x2(-4i + x))(4i + x)
fx = ((24i + -6x) + 1x * (-4i + x) + x2(-4i + x))(4i + x)
fx = (24i + -6x + (-4i * 1x + x * 1x) + x2(-4i + x))(4i + x)
fx = (24i + -6x + (-4ix + 1x2) + x2(-4i + x))(4i + x)
fx = (24i + -6x + -4ix + 1x2 + (-4i * x2 + x * x2))(4i + x)
fx = (24i + -6x + -4ix + 1x2 + (-4ix2 + x3))(4i + x)

Reorder the terms:
fx = (24i + -4ix + -4ix2 + -6x + 1x2 + x3)(4i + x)
fx = (24i + -4ix + -4ix2 + -6x + 1x2 + x3)(4i + x)

Multiply (24i + -4ix + -4ix2 + -6x + 1x2 + x3) * (4i + x)
fx = (24i * (4i + x) + -4ix * (4i + x) + -4ix2 * (4i + x) + -6x * (4i + x) + 1x2 * (4i + x) + x3(4i + x))
fx = ((4i * 24i + x * 24i) + -4ix * (4i + x) + -4ix2 * (4i + x) + -6x * (4i + x) + 1x2 * (4i + x) + x3(4i + x))

Reorder the terms:
fx = ((24ix + 96i2) + -4ix * (4i + x) + -4ix2 * (4i + x) + -6x * (4i + x) + 1x2 * (4i + x) + x3(4i + x))
fx = ((24ix + 96i2) + -4ix * (4i + x) + -4ix2 * (4i + x) + -6x * (4i + x) + 1x2 * (4i + x) + x3(4i + x))
fx = (24ix + 96i2 + (4i * -4ix + x * -4ix) + -4ix2 * (4i + x) + -6x * (4i + x) + 1x2 * (4i + x) + x3(4i + x))

Reorder the terms:
fx = (24ix + 96i2 + (-4ix2 + -16i2x) + -4ix2 * (4i + x) + -6x * (4i + x) + 1x2 * (4i + x) + x3(4i + x))
fx = (24ix + 96i2 + (-4ix2 + -16i2x) + -4ix2 * (4i + x) + -6x * (4i + x) + 1x2 * (4i + x) + x3(4i + x))
fx = (24ix + 96i2 + -4ix2 + -16i2x + (4i * -4ix2 + x * -4ix2) + -6x * (4i + x) + 1x2 * (4i + x) + x3(4i + x))

Reorder the terms:
fx = (24ix + 96i2 + -4ix2 + -16i2x + (-4ix3 + -16i2x2) + -6x * (4i + x) + 1x2 * (4i + x) + x3(4i + x))
fx = (24ix + 96i2 + -4ix2 + -16i2x + (-4ix3 + -16i2x2) + -6x * (4i + x) + 1x2 * (4i + x) + x3(4i + x))
fx = (24ix + 96i2 + -4ix2 + -16i2x + -4ix3 + -16i2x2 + (4i * -6x + x * -6x) + 1x2 * (4i + x) + x3(4i + x))
fx = (24ix + 96i2 + -4ix2 + -16i2x + -4ix3 + -16i2x2 + (-24ix + -6x2) + 1x2 * (4i + x) + x3(4i + x))
fx = (24ix + 96i2 + -4ix2 + -16i2x + -4ix3 + -16i2x2 + -24ix + -6x2 + (4i * 1x2 + x * 1x2) + x3(4i + x))
fx = (24ix + 96i2 + -4ix2 + -16i2x + -4ix3 + -16i2x2 + -24ix + -6x2 + (4ix2 + 1x3) + x3(4i + x))
fx = (24ix + 96i2 + -4ix2 + -16i2x + -4ix3 + -16i2x2 + -24ix + -6x2 + 4ix2 + 1x3 + (4i * x3 + x * x3))
fx = (24ix + 96i2 + -4ix2 + -16i2x + -4ix3 + -16i2x2 + -24ix + -6x2 + 4ix2 + 1x3 + (4ix3 + x4))

Reorder the terms:
fx = (24ix + -24ix + -4ix2 + 4ix2 + -4ix3 + 4ix3 + 96i2 + -16i2x + -16i2x2 + -6x2 + 1x3 + x4)

Combine like terms: 24ix + -24ix = 0
fx = (0 + -4ix2 + 4ix2 + -4ix3 + 4ix3 + 96i2 + -16i2x + -16i2x2 + -6x2 + 1x3 + x4)
fx = (-4ix2 + 4ix2 + -4ix3 + 4ix3 + 96i2 + -16i2x + -16i2x2 + -6x2 + 1x3 + x4)

Combine like terms: -4ix2 + 4ix2 = 0
fx = (0 + -4ix3 + 4ix3 + 96i2 + -16i2x + -16i2x2 + -6x2 + 1x3 + x4)
fx = (-4ix3 + 4ix3 + 96i2 + -16i2x + -16i2x2 + -6x2 + 1x3 + x4)

Combine like terms: -4ix3 + 4ix3 = 0
fx = (0 + 96i2 + -16i2x + -16i2x2 + -6x2 + 1x3 + x4)
fx = (96i2 + -16i2x + -16i2x2 + -6x2 + 1x3 + x4)

Solving
fx = 96i2 + -16i2x + -16i2x2 + -6x2 + 1x3 + x4

Solving for variable 'f'.

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

Divide each side by 'x'.
f = 96i2x-1 + -16i2 + -16i2x + -6x + x2 + x3

Simplifying
f = 96i2x-1 + -16i2 + -16i2x + -6x + x2 + x3

Reorder the terms:
f = -16i2 + 96i2x-1 + -16i2x + -6x + x2 + x3

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