Can $a^2+b+2$ and $b^2+4a$ both be perfect squares? Are there any positive integers $a$ and $b$ so that $a^2+b+2$ and $b^2+4a$ are both perfect squares?

Suppose that $a\le b$. Since $b^2$ is a perfect square, and $b^2+4a\lt (b+2)^2$, $b^2+4a$ cannot be a perfect square unless $b^2+4a=(b+1)^2$, which is impossible.
Now suppose that $a\gt b$. Then $a^2+b+2$ is too small to be a perfect square, since the smallest perfect square greater than $a^2$ is $a^2+2a+1$. 
