2. Given two polynomials $G(x)=x^{4}+A x^{3}+B x^{2}+C x+D$ and $H(x)=x^{2}+A x+B$. Find the value of $G\left(x_{1}\right)+G\left(x_{2}\right)$, where $x_{1}, x_{2}$ are the roots of the polynomial $H(x)$.
Answer: $2 D-A C$.

Solution: Notice that

$$
G(x)=x^{2} \cdot H(x)+C x+D
$$

therefore

$$
G\left(x_{1}\right)+G\left(x_{2}\right)=\left(x_{1}^{2} \cdot H\left(x_{1}\right)+C x_{1}+D\right)+\left(x_{2}^{2} \cdot H\left(x_{2}\right)+C x_{2}+D\right)=C\left(x_{1}+x_{2}\right)+2 D
$$

It remains to notice that by Vieta's theorem $x_{1}+x_{2}=-A$. Therefore,

$$
G\left(x_{1}\right)+G\left(x_{2}\right)=C\left(x_{1}+c_{2}\right)+2 D=2 D-A C
$$

Answers:

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