Find the Partial Fraction Decomposition

Question

$\frac{2x^5+3x^4-3x^3-2x^2+x}{2x^2+5x+2}$, I am not sure as to where to start with this one; I have already done the factoring process of the denominator but not sure how to continue the algebraic process. Solve and leave steps listed, please.

Answer 1

If you perform long division you should obtain:

$$x^3-x^2+\frac{x}{2x^2+5x+2}\; .$$

Then, the denominator can be factored to give

$$x^3-x^2+\frac{x}{(x+2)(2x+1)}\; .$$

And this can be broken up into partial fractions

$$x^3-x^2+\frac{2}{3(x+2)}-\frac{1}{3(2x+1)}\; .$$