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

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

Solve for B

$$B=-\frac{-x^{3}+x^{2}-5x-1}{2x-3}$$

$$x\neq \frac{3}{2}$$

Steps for Solving Linear Equation

Use the distributive property to multiply B by 2x-3.

$$x^{3}-x^{2}+3x+1=2Bx-3B-2x$$

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

$$2Bx-3B-2x=x^{3}-x^{2}+3x+1$$

Add 2x to both sides.

$$2Bx-3B=x^{3}-x^{2}+3x+1+2x$$

Combine 3x and 2x to get 5x.

$$2Bx-3B=x^{3}-x^{2}+5x+1$$

Combine all terms containing B.

$$\left(2x-3\right)B=x^{3}-x^{2}+5x+1$$

Divide both sides by 2x-3.

$$\frac{\left(2x-3\right)B}{2x-3}=\frac{x^{3}-x^{2}+5x+1}{2x-3}$$

Dividing by 2x-3 undoes the multiplication by 2x-3.

$$B=\frac{x^{3}-x^{2}+5x+1}{2x-3}$$