Comparing coefficients with polynomials

Question

$x^5-x^4-2x^3+2x^2-3$ is identical to $(x+1)(x-2)Q(x)+ax-b$, where $Q(x)$ is a polynomial

Answer 1

By degree reasons $Q(x)$ is a monic polynomial of degree $3$. Comparison of coefficients gives $$ Q(x)=x^3 +2, $$ which in turn determine $a$ and $b$, namely $a=2$ and $b=-1$.

Answer 2

Since the formula $$x^5-x^4-2x^3+2x^2-3=(x+1)(x-2)Q(x)+ax-b$$ is valid for all $x$ it is valid also for

$$x=2:\;\;\;\; 32-16-16+8-3 =0+2a-b \implies \boxed{2a-b=5} $$ and

$$x=-1:\;\;\;\; -1-1+2+2-3 =0-a-b\implies \boxed{a+b=1}$$

Now solve this system and you are done.