Solution:
As $a b = c d$, we get $a(a + b + c + d) = (a + c)(a + d)$. If $a + b + c + d$ were a prime, then it would be a factor in either $a + c$ or $a + d$, which are both smaller than $a + b + c + d$.

Solution:
Let $r = \operatorname{gcd}(a, c)$ and $s = \operatorname{gcd}(b, d)$. Let $a = a' r$, $b = b' s$, $c = c' r$ and $d = d' s$. Then $a' b' = c' d'$. But $\operatorname{gcd}(a', c') = 1$ and $\operatorname{gcd}(b', d') = 1$, so we must have $a' = d'$ and $b' = c'$. This gives
$$a + b + c + d = a' r + b' s + c' r + d' s = a' r + b' s + b' r + a' s = (a' + b')(r + s).$$
Since $a'$, $b'$, $r$ and $s$ are positive integers, $a + b + c + d$ is not a prime.