Ideal class groups of certain real quadratic number fields and factorizations of values of certain quadratic polynomials

We determine the odd primes $p$ for which the number of distinct prime divisors of $d-x^2$ is at most $2$ for every positive odd integers $x\le \sqrt{p}$. We also determine the odd primes $p$ for which the number of distinct prime divisors of $2p-x^2$ is at most $2$ for every positive even integers $x\le \sqrt{2p}$. To this end, we show that these properties of $p$ and $2p$ impose restrictions on the class numbers and fundamental units of the real quadratic fields $\mathbb{Q}(\sqrt{p})$ and $\mathbb{Q}(\sqrt{2p})$.