I have a function $f(x) = 4x^2 + 4ax + b$ where $a$ and $b$ are both even numbers. I also have an interval within which there are only two integers $x$ that will result in a perfect square. $a$, $b$, and the larger of the two perfect squares are known.
Is there some way that does not involve prime factorization to find or get a rough estimate of what value $x$ will result in the smaller perfect square?
Let us call these two values of $x, \ x_1$ and $x_2$, being the larger of the two. Then as $x_2$ is known, we can write that $x_1=x_2-m$ for some integer $m$. Then we get that we know $f(x_1)$ to be a perfect square we get that $f(x_2-m)$ is also a perfect square. $$f(x_2-m)=4(x_2-m)^2+4a(x_2-m)+b=4m^2-(8x_2+4a)m+(4x_2^2+4ax_2+b)$$ which is a quadratic in $m$. Solve this for $m$ to be in your known interval and you have $x_1$