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

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


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

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

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

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

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

Multiply (-1 + x) * (-4i + x)
fx = a(-1(-4i + x) + x(-4i + x))(4i + x)
fx = a((-4i * -1 + x * -1) + x(-4i + x))(4i + x)
fx = a((4i + -1x) + x(-4i + x))(4i + x)
fx = a(4i + -1x + (-4i * x + x * x))(4i + x)
fx = a(4i + -1x + (-4ix + x2))(4i + x)

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

Multiply (4i + -4ix + -1x + x2) * (4i + x)
fx = a(4i * (4i + x) + -4ix * (4i + x) + -1x * (4i + x) + x2(4i + x))
fx = a((4i * 4i + x * 4i) + -4ix * (4i + x) + -1x * (4i + x) + x2(4i + x))

Reorder the terms:
fx = a((4ix + 16i2) + -4ix * (4i + x) + -1x * (4i + x) + x2(4i + x))
fx = a((4ix + 16i2) + -4ix * (4i + x) + -1x * (4i + x) + x2(4i + x))
fx = a(4ix + 16i2 + (4i * -4ix + x * -4ix) + -1x * (4i + x) + x2(4i + x))

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

Reorder the terms:
fx = a(4ix + -4ix + -4ix2 + 4ix2 + 16i2 + -16i2x + -1x2 + x3)

Combine like terms: 4ix + -4ix = 0
fx = a(0 + -4ix2 + 4ix2 + 16i2 + -16i2x + -1x2 + x3)
fx = a(-4ix2 + 4ix2 + 16i2 + -16i2x + -1x2 + x3)

Combine like terms: -4ix2 + 4ix2 = 0
fx = a(0 + 16i2 + -16i2x + -1x2 + x3)
fx = a(16i2 + -16i2x + -1x2 + x3)
fx = (16i2 * a + -16i2x * a + -1x2 * a + x3 * a)
fx = (16ai2 + -16ai2x + -1ax2 + ax3)

Solving
fx = 16ai2 + -16ai2x + -1ax2 + 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 = 16ai2x-1 + -16ai2 + -1ax + ax2

Simplifying
f = 16ai2x-1 + -16ai2 + -1ax + ax2

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

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