Consider the polynomial $$P = x^2 + 2(a-1)x - 8a - 8.$$ (1) Let $a$ be a rational number. If the value of $P$ is a rational number when $x = 1 - \sqrt{2}$, then $a =$ $\mathbf{K}$ and in this case the value of $P$ is $P =$ $\mathbf{LM}$. (2) Let $x$ and $a$ be positive integers. We are to investigate $x$ and $a$ which are such that the value of $P$ is a prime number. When we factorize $P$, we have $$P = (x - \mathbf{N})(x + \mathbf{O}a + \mathbf{P}).$$ Hence $x$ must be $\mathbf{Q}$. Furthermore, the smallest possible $a$ is $\mathbf{R}$, and in this case the value of $P$ is $P = \mathbf{ST}$.
Consider the polynomial
$$P = x^2 + 2(a-1)x - 8a - 8.$$
(1) Let $a$ be a rational number. If the value of $P$ is a rational number when $x = 1 - \sqrt{2}$, then $a =$ $\mathbf{K}$ and in this case the value of $P$ is $P =$ $\mathbf{LM}$.
(2) Let $x$ and $a$ be positive integers. We are to investigate $x$ and $a$ which are such that the value of $P$ is a prime number.
When we factorize $P$, we have
$$P = (x - \mathbf{N})(x + \mathbf{O}a + \mathbf{P}).$$
Hence $x$ must be $\mathbf{Q}$.
Furthermore, the smallest possible $a$ is $\mathbf{R}$, and in this case the value of $P$ is $P = \mathbf{ST}$.