When factoring polynomials, a good strategy is to use the Rational Root Theorem. When applying this theorem, it is a good idea to try substituting +-1 into the polynomial and seeing if the result is 0 because these substitutions are easy and can be done quickly. If +-1 doesn't give 0, try substituting in other factors of the constant term divided by factors of the leading term.
Looking at p^4+2p^3+2p^2-2p-3, trying p=1 gives 0 so we know p-1 is a factor. Using synthetic division (or polynomial long division), we see that p^4+2p^3+2p^2-2p-3=(p-1)(p^3+3p^2+5p+3). Now trying p=-1 into the cubic polynomial gives 0 so we get (p-1)(p^3+3p^2+5p+3)=(p-1)(p+1)(p^2+2p+3). Looking at the quadratic polynomial, we know it is irreducible because no factors of 3 add up to 2 and we are done.
