The left-hand side is always even for any integers $x, y, z$, while the right-hand side is odd.

The greatest common divisor of the coefficients on the left-hand side, $\gcd(2, 2, 4) = 2$, does not divide the right-hand side $5$.

There are no positive values for $x, y, z$ that can sum to $5$.

The equation has infinitely many rational solutions but no integer solutions because $5$ is a prime number.