Solve 2x^3+(1-y)x+y-3=(x-1)(2x^2+(3-y)x+3)

$$2 { x }^{ 3 } +(1-y)x+y-3=(x-1)(2 { x }^{ 2 } +(3-y)x+3)$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=\frac{x}{x-1}\text{, }&x\neq 1\\y\in \mathrm{C}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 1-y by x.

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

Use the distributive property to multiply 3-y by x.

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

Use the distributive property to multiply x-1 by 2x^{2}+3x-yx+3 and combine like terms.

$$2x^{3}+x-yx+y-3=2x^{3}+x^{2}-yx^{2}+yx-3$$

Add yx^{2} to both sides.

$$2x^{3}+x-yx+y-3+yx^{2}=2x^{3}+x^{2}+yx-3$$

Subtract yx from both sides.

$$2x^{3}+x-yx+y-3+yx^{2}-yx=2x^{3}+x^{2}-3$$

Combine -yx and -yx to get -2yx.

$$2x^{3}+x-2yx+y-3+yx^{2}=2x^{3}+x^{2}-3$$

Subtract 2x^{3} from both sides.

$$x-2yx+y-3+yx^{2}=2x^{3}+x^{2}-3-2x^{3}$$

Combine 2x^{3} and -2x^{3} to get 0.

$$x-2yx+y-3+yx^{2}=x^{2}-3$$

Subtract x from both sides.

$$-2yx+y-3+yx^{2}=x^{2}-3-x$$

Add 3 to both sides.

$$-2yx+y+yx^{2}=x^{2}-3-x+3$$

Add -3 and 3 to get 0.

$$-2yx+y+yx^{2}=x^{2}-x$$

Combine all terms containing y.

$$\left(-2x+1+x^{2}\right)y=x^{2}-x$$

The equation is in standard form.

$$\left(x^{2}-2x+1\right)y=x^{2}-x$$

Divide both sides by -2x+1+x^{2}.

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

Dividing by -2x+1+x^{2} undoes the multiplication by -2x+1+x^{2}.

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

Divide x\left(-1+x\right) by -2x+1+x^{2}.

$$y=\frac{x}{x-1}$$

Solve for y

$$\left\{\begin{matrix}y=\frac{x}{x-1}\text{, }&x\neq 1\\y\in \mathrm{R}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 1-y by x.

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

Use the distributive property to multiply 3-y by x.

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

Use the distributive property to multiply x-1 by 2x^{2}+3x-yx+3 and combine like terms.

$$2x^{3}+x-yx+y-3=2x^{3}+x^{2}-yx^{2}+yx-3$$

Add yx^{2} to both sides.

$$2x^{3}+x-yx+y-3+yx^{2}=2x^{3}+x^{2}+yx-3$$

Subtract yx from both sides.

$$2x^{3}+x-yx+y-3+yx^{2}-yx=2x^{3}+x^{2}-3$$

Combine -yx and -yx to get -2yx.

$$2x^{3}+x-2yx+y-3+yx^{2}=2x^{3}+x^{2}-3$$

Subtract 2x^{3} from both sides.

$$x-2yx+y-3+yx^{2}=2x^{3}+x^{2}-3-2x^{3}$$

Combine 2x^{3} and -2x^{3} to get 0.

$$x-2yx+y-3+yx^{2}=x^{2}-3$$

Subtract x from both sides.

$$-2yx+y-3+yx^{2}=x^{2}-3-x$$

Add 3 to both sides.

$$-2yx+y+yx^{2}=x^{2}-3-x+3$$

Add -3 and 3 to get 0.

$$-2yx+y+yx^{2}=x^{2}-x$$

Combine all terms containing y.

$$\left(-2x+1+x^{2}\right)y=x^{2}-x$$

The equation is in standard form.

$$\left(x^{2}-2x+1\right)y=x^{2}-x$$

Divide both sides by -2x+1+x^{2}.

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

Dividing by -2x+1+x^{2} undoes the multiplication by -2x+1+x^{2}.

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

Divide x\left(-1+x\right) by -2x+1+x^{2}.

$$y=\frac{x}{x-1}$$

Solve for x

$$\left\{\begin{matrix}\\x=1\text{, }&\text{unconditionally}\\x=\frac{y}{y-1}\text{, }&y\neq 1\end{matrix}\right.$$