1. Real numbers $a$ and $b$ satisfy that the quadratic trinomials $x^{2}+a x+b$ and $x^{2}+b x+a$ both have two distinct real roots, and their product has exactly three distinct real roots. Find all possible values of the sum of these three distinct roots.
1. Given the condition, let's assume the quadratic trinomials $x^{2}+a x+b$ and $x^{2}+b x+a$ have exactly one common real root $x_{0}$.
Then $a \neq b$,
$$
\begin{array}{l}
0=\left(x^{2}+a x+b\right)-\left(x^{2}+b x+a\right) \\
=(a-b)\left(x_{0}-1\right) .
\end{array}
$$

Therefore, $x_{0}=1, a+b+1=0$.
By Vieta's formulas, the sum of the three distinct real roots is
$$
-a-b-x_{0}=0 .
$$