Solve (x-1)^2tp(x)=x^4-2x^3+3x-2

$$( x - 1 ) ^ { 2 } t p ( x ) = x ^ { 4 } - 2 x ^ { 3 } + 3 x - 2$$

Solve for p

$$\left\{\begin{matrix}p=-\frac{-x^{3}+x^{2}+x-2}{tx\left(x-1\right)}\text{, }&t\neq 0\text{ and }x\neq 1\text{ and }x\neq 0\\p\in \mathrm{R}\text{, }&\left(x=-\sqrt[3]{\frac{\sqrt{177}}{18}+\frac{43}{54}}-\sqrt[3]{-\frac{\sqrt{177}}{18}+\frac{43}{54}}+\frac{1}{3}\text{ and }t=0\right)\text{ or }x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$\left(x^{2}-2x+1\right)tpx=x^{4}-2x^{3}+3x-2$$

Use the distributive property to multiply x^{2}-2x+1 by t.

$$\left(x^{2}t-2xt+t\right)px=x^{4}-2x^{3}+3x-2$$

Use the distributive property to multiply x^{2}t-2xt+t by p.

$$\left(x^{2}tp-2xtp+tp\right)x=x^{4}-2x^{3}+3x-2$$

Use the distributive property to multiply x^{2}tp-2xtp+tp by x.

$$tpx^{3}-2tpx^{2}+tpx=x^{4}-2x^{3}+3x-2$$

Combine all terms containing p.

$$\left(tx^{3}-2tx^{2}+tx\right)p=x^{4}-2x^{3}+3x-2$$

Divide both sides by tx^{3}-2tx^{2}+tx.

$$\frac{\left(tx^{3}-2tx^{2}+tx\right)p}{tx^{3}-2tx^{2}+tx}=\frac{x^{4}-2x^{3}+3x-2}{tx^{3}-2tx^{2}+tx}$$

Dividing by tx^{3}-2tx^{2}+tx undoes the multiplication by tx^{3}-2tx^{2}+tx.

$$p=\frac{x^{4}-2x^{3}+3x-2}{tx^{3}-2tx^{2}+tx}$$

Divide x^{4}-2x^{3}+3x-2 by tx^{3}-2tx^{2}+tx.

$$p=\frac{x^{3}-x^{2}-x+2}{tx\left(x-1\right)}$$

Solve for t

$$\left\{\begin{matrix}t=-\frac{-x^{3}+x^{2}+x-2}{px\left(x-1\right)}\text{, }&p\neq 0\text{ and }x\neq 1\text{ and }x\neq 0\\t\in \mathrm{R}\text{, }&\left(x=-\sqrt[3]{\frac{\sqrt{177}}{18}+\frac{43}{54}}-\sqrt[3]{-\frac{\sqrt{177}}{18}+\frac{43}{54}}+\frac{1}{3}\text{ and }p=0\right)\text{ or }x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$\left(x^{2}-2x+1\right)tpx=x^{4}-2x^{3}+3x-2$$

Use the distributive property to multiply x^{2}-2x+1 by t.

$$\left(x^{2}t-2xt+t\right)px=x^{4}-2x^{3}+3x-2$$

Use the distributive property to multiply x^{2}t-2xt+t by p.

$$\left(x^{2}tp-2xtp+tp\right)x=x^{4}-2x^{3}+3x-2$$

Use the distributive property to multiply x^{2}tp-2xtp+tp by x.

$$tpx^{3}-2tpx^{2}+tpx=x^{4}-2x^{3}+3x-2$$

Combine all terms containing t.

$$\left(px^{3}-2px^{2}+px\right)t=x^{4}-2x^{3}+3x-2$$

Divide both sides by px^{3}-2px^{2}+px.

$$\frac{\left(px^{3}-2px^{2}+px\right)t}{px^{3}-2px^{2}+px}=\frac{x^{4}-2x^{3}+3x-2}{px^{3}-2px^{2}+px}$$

Dividing by px^{3}-2px^{2}+px undoes the multiplication by px^{3}-2px^{2}+px.

$$t=\frac{x^{4}-2x^{3}+3x-2}{px^{3}-2px^{2}+px}$$

Divide x^{4}-2x^{3}+3x-2 by px^{3}-2px^{2}+px.

$$t=\frac{x^{3}-x^{2}-x+2}{px\left(x-1\right)}$$