help me people

My math teacher sent me some homework via EMAIL at YESTERDAY MIDNIGHT. And he wrote that me and another student has to solve all the tasks, or else we will be given enough grades to fail at the end of this term. ISN'T HE CRAZY???

So please, my fellows, help me shove the answers up his ass.
Here is the first i couldn't solve.

x^2 + 32x + 2264 = p^2; p is a prime of choice
find x, if x is an integer

tl;dr: please give tips how to solve the problem above

>>1
I shall halp.

In either case, let y = x+16.  Then we can just solve for y and subtract 16 to get x.

y^2 + 2008 = p^2

(p-y)(p+y) = 2008

2008 = 8 * 251 (251 is prime)

p-y and p+y are both odd or both even.  Since you can't write 2008 as a product of odd numbers, they must both be even.  Therefore (p-y, p+y) must be either
(2, 4 * 251)
(4, 2 * 251)

If (a,b) is one of these pairs, then
p-y = a
p+y = b
and so p = (a+b)/2.

(2+4*251)/2 = 503 is prime, which gives the solution p=503, y=501 (and x=501-16=485).  However (4+2*251)/2 = 253 = 11*23 is not.  Therefore the only solution is

p = 503
x = 485