Let $n$ be a natural number and $p$ a prime number less than or equal to $n$.
$$\begin{align} n^2 + 2n &\equiv a \pmod p\\ n^2 + 1 &\equiv b \pmod p \end{align}$$
If $a \lt b$, $p$ is an answer for the equation.
Is there an upper bound for the number of answers to this equation?
I checked some cases and guess for $n\ge k$, an upper bound for the number of answers is $\frac t2$.
Let $t$ be the number of primes less than or equal to $n$.
For example consider $n = 20$:
All prime numbers less than or equal to $20$ are $2,3,5,7,11,13,17,19$
$$\begin{align} 20^2 + 2(20) &= 0 \pmod 2\\ 20^2 + 1 &= 1 \pmod 2\\[5pt] 0 &< 1 \end{align}$$
…so $p=2$ is an answer of the equation.