Factorize the polynomial $P(X)=2 X^{3}+3 X^{2}-3 X-2$ in $\mathbb{R}$.
Here, to reduce to a case we know well (polynomial of degree 2), we try to find an "obvious root." We then quickly calculate $P(1), P(-1)$ to see if the polynomial $P$ vanishes at $\pm 1$ (but we rarely try more values by hand...)

In this case, we notice that $P(1)=0$, so we can rewrite $P: P(X)=(X-1) Q(X)$ with $Q(X)=2 X^{2}+5 X+2$.

We then calculate the discriminant: $\Delta=5^{2}-4 \times 2 \times 2=25-16=9>0$.

The roots of $Q$ are then $x_{1}=\frac{-5+3}{4}=\frac{-1}{2}$ and $x_{2}=\frac{-5-3}{4}=-2$.

We can therefore write $P=2(X-1)(X+2)\left(X+\frac{1}{2}\right)$.