## Task B-1.1.

Determine all integers $a$ for which $4 a^{2}-24 a-45$ is a prime number.
## Solution.

Let $4 a^{2}-24 a-45=p$, where $p$ is a prime number. Factorize the expression on the left side of the equation.

$$
\begin{aligned}
4 a^{2}-24 a-45 & =4 a^{2}-30 a+6 a-45 \\
& =2 a(2 a-15)+3(2 a-15) \\
& =(2 a+3)(2 a-15)
\end{aligned}
$$

Then from $(2 a+3)(2 a-15)=p$ it follows:

1) $2 a+3= \pm 1$ and $2 a-15= \pm p \Longrightarrow a=-1, p=-17$ or $a=-2, p=19 . \quad 1$ point

Since $p>0$, in this case the only possibility is $a=-2, p=19 . \quad 1$ point
2) $2 a+3= \pm p$ and $2 a-15= \pm 1 \Longrightarrow a=8, p=19$ or $a=7, p=-17 . \quad 1$ point

In this case, the only possibility is $a=8, p=19$.

Therefore, the only integers for which the given expression is a prime number are -2 and 8. 1 point