f(x)*g(x)=(x+6)(4x+6)

Solution for f(x)*g(x)=(x+6)(4x+6) equation:

Simplifying
f(x) * g(x) = (x + 6)(4x + 6)

Multiply f * x
fx * g * x = (x + 6)(4x + 6)

Multiply fx * g
fgx * x = (x + 6)(4x + 6)

Multiply fgx * x
fgx2 = (x + 6)(4x + 6)

Reorder the terms:
fgx2 = (6 + x)(4x + 6)

Reorder the terms:
fgx2 = (6 + x)(6 + 4x)

Multiply (6 + x) * (6 + 4x)
fgx2 = (6(6 + 4x) + x(6 + 4x))
fgx2 = ((6 * 6 + 4x * 6) + x(6 + 4x))
fgx2 = ((36 + 24x) + x(6 + 4x))
fgx2 = (36 + 24x + (6 * x + 4x * x))
fgx2 = (36 + 24x + (6x + 4x2))

Combine like terms: 24x + 6x = 30x
fgx2 = (36 + 30x + 4x2)

Solving
fgx2 = 36 + 30x + 4x2

Solving for variable 'f'.

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

Divide each side by 'gx2'.
f = 36g-1x-2 + 30g-1x-1 + 4g-1

Simplifying
f = 36g-1x-2 + 30g-1x-1 + 4g-1

Reorder the terms:
f = 4g-1 + 36g-1x-2 + 30g-1x-1