$a,b,c,d$ are positive integers such that $ad=bc$. Prove that $n=a+b+c+d$ cannot be prime

I'm not sure how to end your solution, but I have another one if you are interested:

Suppose $a+b+c+d=p\in \mathbb{P}$. Since $d=p-a-b-c$ we get $$a(p-a-b-c)=bc$$ so $$ ap = (a+b)(a+c) \implies p\mid a+b \;\;\;\;{\rm or }\;\;\;\;p\mid a+c$$

A contradiction (since $p>a+b$ and $p>a+c$).

Since $d = bc/a$ is an integer, we can factor $d = (b/x)(c/y)$ for some positive integers $x,y$ such that $b/x$ and $c/y$ are integers and $xy=a$. (This is intuitively clear and can be made rigorous by playing around with prime factorizations of $a,b,c$.)

Then $$ a + b + c + d = xy + b + c + (b/x)(c/y) = (x + c/y)(y + b/x) $$ which is always a nontrivial factorization, so $a+b+c+d$ cannot be prime.