A cybersecurity researcher is analyzing a specific encryption key property. They need to prove that the equation $p^2-4q=2$ has no integer solutions for $p$ and $q$.

The researcher starts a proof by contradiction as shown below:

"Suppose there exist integers $p$ and $q$ such that $p^2-4q=2$. This implies that

$$p^2=4q+2$$

so $p^2$ is even."

Complete the proof.