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

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


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

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

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

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

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

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

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

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

Multiply (4 + -4x + x2) * (3i + x)
fx = (4(3i + x) + -4x * (3i + x) + x2(3i + x))(-3i + x)
fx = ((3i * 4 + x * 4) + -4x * (3i + x) + x2(3i + x))(-3i + x)
fx = ((12i + 4x) + -4x * (3i + x) + x2(3i + x))(-3i + x)
fx = (12i + 4x + (3i * -4x + x * -4x) + x2(3i + x))(-3i + x)
fx = (12i + 4x + (-12ix + -4x2) + x2(3i + x))(-3i + x)
fx = (12i + 4x + -12ix + -4x2 + (3i * x2 + x * x2))(-3i + x)
fx = (12i + 4x + -12ix + -4x2 + (3ix2 + x3))(-3i + x)

Reorder the terms:
fx = (12i + -12ix + 3ix2 + 4x + -4x2 + x3)(-3i + x)
fx = (12i + -12ix + 3ix2 + 4x + -4x2 + x3)(-3i + x)

Multiply (12i + -12ix + 3ix2 + 4x + -4x2 + x3) * (-3i + x)
fx = (12i * (-3i + x) + -12ix * (-3i + x) + 3ix2 * (-3i + x) + 4x * (-3i + x) + -4x2 * (-3i + x) + x3(-3i + x))
fx = ((-3i * 12i + x * 12i) + -12ix * (-3i + x) + 3ix2 * (-3i + x) + 4x * (-3i + x) + -4x2 * (-3i + x) + x3(-3i + x))

Reorder the terms:
fx = ((12ix + -36i2) + -12ix * (-3i + x) + 3ix2 * (-3i + x) + 4x * (-3i + x) + -4x2 * (-3i + x) + x3(-3i + x))
fx = ((12ix + -36i2) + -12ix * (-3i + x) + 3ix2 * (-3i + x) + 4x * (-3i + x) + -4x2 * (-3i + x) + x3(-3i + x))
fx = (12ix + -36i2 + (-3i * -12ix + x * -12ix) + 3ix2 * (-3i + x) + 4x * (-3i + x) + -4x2 * (-3i + x) + x3(-3i + x))

Reorder the terms:
fx = (12ix + -36i2 + (-12ix2 + 36i2x) + 3ix2 * (-3i + x) + 4x * (-3i + x) + -4x2 * (-3i + x) + x3(-3i + x))
fx = (12ix + -36i2 + (-12ix2 + 36i2x) + 3ix2 * (-3i + x) + 4x * (-3i + x) + -4x2 * (-3i + x) + x3(-3i + x))
fx = (12ix + -36i2 + -12ix2 + 36i2x + (-3i * 3ix2 + x * 3ix2) + 4x * (-3i + x) + -4x2 * (-3i + x) + x3(-3i + x))

Reorder the terms:
fx = (12ix + -36i2 + -12ix2 + 36i2x + (3ix3 + -9i2x2) + 4x * (-3i + x) + -4x2 * (-3i + x) + x3(-3i + x))
fx = (12ix + -36i2 + -12ix2 + 36i2x + (3ix3 + -9i2x2) + 4x * (-3i + x) + -4x2 * (-3i + x) + x3(-3i + x))
fx = (12ix + -36i2 + -12ix2 + 36i2x + 3ix3 + -9i2x2 + (-3i * 4x + x * 4x) + -4x2 * (-3i + x) + x3(-3i + x))
fx = (12ix + -36i2 + -12ix2 + 36i2x + 3ix3 + -9i2x2 + (-12ix + 4x2) + -4x2 * (-3i + x) + x3(-3i + x))
fx = (12ix + -36i2 + -12ix2 + 36i2x + 3ix3 + -9i2x2 + -12ix + 4x2 + (-3i * -4x2 + x * -4x2) + x3(-3i + x))
fx = (12ix + -36i2 + -12ix2 + 36i2x + 3ix3 + -9i2x2 + -12ix + 4x2 + (12ix2 + -4x3) + x3(-3i + x))
fx = (12ix + -36i2 + -12ix2 + 36i2x + 3ix3 + -9i2x2 + -12ix + 4x2 + 12ix2 + -4x3 + (-3i * x3 + x * x3))
fx = (12ix + -36i2 + -12ix2 + 36i2x + 3ix3 + -9i2x2 + -12ix + 4x2 + 12ix2 + -4x3 + (-3ix3 + x4))

Reorder the terms:
fx = (12ix + -12ix + -12ix2 + 12ix2 + 3ix3 + -3ix3 + -36i2 + 36i2x + -9i2x2 + 4x2 + -4x3 + x4)

Combine like terms: 12ix + -12ix = 0
fx = (0 + -12ix2 + 12ix2 + 3ix3 + -3ix3 + -36i2 + 36i2x + -9i2x2 + 4x2 + -4x3 + x4)
fx = (-12ix2 + 12ix2 + 3ix3 + -3ix3 + -36i2 + 36i2x + -9i2x2 + 4x2 + -4x3 + x4)

Combine like terms: -12ix2 + 12ix2 = 0
fx = (0 + 3ix3 + -3ix3 + -36i2 + 36i2x + -9i2x2 + 4x2 + -4x3 + x4)
fx = (3ix3 + -3ix3 + -36i2 + 36i2x + -9i2x2 + 4x2 + -4x3 + x4)

Combine like terms: 3ix3 + -3ix3 = 0
fx = (0 + -36i2 + 36i2x + -9i2x2 + 4x2 + -4x3 + x4)
fx = (-36i2 + 36i2x + -9i2x2 + 4x2 + -4x3 + x4)

Solving
fx = -36i2 + 36i2x + -9i2x2 + 4x2 + -4x3 + 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 = -36i2x-1 + 36i2 + -9i2x + 4x + -4x2 + x3

Simplifying
f = -36i2x-1 + 36i2 + -9i2x + 4x + -4x2 + x3

Reorder the terms:
f = 36i2 + -36i2x-1 + -9i2x + 4x + -4x2 + x3

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