Solve x(x-1)^4/4(1-x^2)=f(x)

$$\frac { x ( x - 1 ) ^ { 4 } } { 4 ( 1 - x ^ { 2 } ) } = f ( x )$$

Solve for f

$$\left\{\begin{matrix}f=-\frac{\left(x-1\right)^{3}}{4\left(x+1\right)}\text{, }&|x|\neq 1\\f\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4\left(x-1\right)\left(-x-1\right).

$$x\left(x-1\right)^{4}=fx\times 4\left(x-1\right)\left(-x-1\right)$$

Use the distributive property to multiply fx\times 4 by x-1.

$$x\left(x-1\right)^{4}=\left(4fx^{2}-fx\times 4\right)\left(-x-1\right)$$

Multiply -1 and 4 to get -4.

$$x\left(x-1\right)^{4}=\left(4fx^{2}-4fx\right)\left(-x-1\right)$$

Use the distributive property to multiply 4fx^{2}-4fx by -x-1 and combine like terms.

$$x\left(x-1\right)^{4}=-4x^{3}f+4xf$$

Swap sides so that all variable terms are on the left hand side.

$$-4x^{3}f+4xf=x\left(x-1\right)^{4}$$

Combine all terms containing f.

$$\left(-4x^{3}+4x\right)f=x\left(x-1\right)^{4}$$

The equation is in standard form.

$$\left(4x-4x^{3}\right)f=x\left(x-1\right)^{4}$$

Divide both sides by -4x^{3}+4x.

$$\frac{\left(4x-4x^{3}\right)f}{4x-4x^{3}}=\frac{x\left(x-1\right)^{4}}{4x-4x^{3}}$$

Dividing by -4x^{3}+4x undoes the multiplication by -4x^{3}+4x.

$$f=\frac{x\left(x-1\right)^{4}}{4x-4x^{3}}$$

Divide x\left(x-1\right)^{4} by -4x^{3}+4x.

$$f=-\frac{\left(x-1\right)^{3}}{4\left(x+1\right)}$$