13.343. It is conjectured that the expression

$$
(x+a)(x+2 a)(x+3 a)(x+4 a)+a^{4}
$$

is the square of a quadratic polynomial of the form $x^{2}+p x+q a^{2}$. How can this statement be verified and the coefficients $p$ and $q$ be found?
Solution.

To verify the given statement, we equate the expressions:

$$
(x+a)(x+2a)(x+3a)(x+4a)+a^{4}=\left(x^{2}+p x+q a^{2}\right)^{2}
$$

expand the brackets

$$
\begin{aligned}
& x^{4}+10 a x^{3}+35 a^{2} x^{2}+50 a^{3} x+25 a^{4}=x^{4}+2 p x^{3}+\left(2 q a^{2}+p^{2}\right) x^{2}+ \\
& +2 p q a^{2} x+q^{2} a^{4}
\end{aligned}
$$

we check if the coefficients of \(x^{4}, x^{3}, x^{2}, x\) and the constant terms \(25 a^{4}\) and \(q^{2} a^{4}\) are equal. If the statement is true, then \(10 a=2 p\), from which \(p=5 a\), and \(35 a^{2}=2 q a^{2}+p^{2}\), from which \(q=5\). We then verify that for the found values of \(p\) and \(q\), \(50 a^{3}=2 p q a^{2}\) and \(25 a^{4}=q^{2} a^{4}\) hold true.

Answer: \(p=5 a ; q=5\).