306. Let $\alpha$ and $\beta$ be the roots of the equation

$$
x^{2} + p x + q = 0
$$

and let $\gamma$ and $\delta$ be the roots of the equation

$$
x^{2} + P x + Q = 0
$$

Express the product

$$
(\alpha-\gamma)(\beta-\gamma) \quad, (\alpha-\delta)(\beta-\delta)
$$

in terms of the coefficients of the given equations.
306. If $\alpha$ and $\beta$ are the roots of the equation

$$
x^{2}+p x+q=0
$$

then

$$
(x-\alpha)(x-\beta)=x^{2}+p x+q
$$

Therefore,

$$
\begin{aligned}
& (\alpha-\gamma)(\beta-\gamma)(\alpha-\delta)(\beta-\delta)= \\
& =[(\gamma-\alpha)(\gamma-\beta)][(\delta-\alpha)(\delta-\beta)]=\left(\gamma^{2}+p \gamma+q\right)\left(\delta^{2}+p \delta+q\right) \\
& \gamma+\delta=-P, \quad \gamma \delta=Q
\end{aligned}
$$

and, thus,

$$
\begin{aligned}
& (\alpha-\gamma)(\beta-\gamma)(\alpha-\delta)(\beta-\delta)=\left(\gamma^{2}+p \gamma+q\right)\left(\delta^{2}+p \delta+q\right)= \\
& =\gamma^{2} \delta^{2}+p \gamma^{2} \delta+q \gamma^{2}+p \gamma \delta^{2}+p^{2} \gamma \delta+p q \gamma+q \delta^{2}+p q \delta+q^{2}= \\
& =(\gamma \delta)^{2}+p \gamma \delta(\gamma+\delta)+q\left[(\gamma+\delta)^{2}-2 \gamma \delta\right]+p^{2} \gamma \delta+p q(\gamma+\delta)+q^{2}= \\
& =Q^{2}-p F Q+q\left(F^{2}-2 Q\right)+p^{2} Q-p q P+q^{2}= \\
& =Q^{2}+q^{2}-p P(Q+q)+q P^{2}+p^{2} Q-2 q Q .
\end{aligned}
$$