Let $a$ and $b$ be two distinct roots of the polynomial $X^{3}+3 X^{2}+X+1$. Calculate $a^{2} b+a b^{2}+3 a b$.
Let $c$ be the third root of this polynomial. The relations between coefficients and roots give:

$$
\left\{\begin{aligned}
\sigma_{1}=a+b+c & =-3 \\
\sigma_{2}=a b+b c+c a & =1 \\
\sigma_{3}=a b c \quad & =-1
\end{aligned}\right.
$$

Thus $a^{2} b+a b^{2}+3 a b=a^{2} b+a b^{2}-(a+b+c) a b=-a b c=1$.