To add the polynomial expressions \(3 - 2p - 5p^2\) and \(p^4 - 3p + 4\), we combine like terms.
First, write the two polynomials together:
\[ (3 - 2p - 5p^2) + (p^4 - 3p + 4) \]
Now, group the like terms:
- The \(p^4\) term: \(p^4\)
- The \(p^2\) term: \(-5p^2\)
- The \(p\) terms: \(-2p - 3p = -5p\)
- The constant terms: \(3 + 4 = 7\)
Now we can combine them:
\[ p^4 - 5p^2 - 5p + 7 \]
Thus, the sum of the two polynomial expressions is:
\[ \boxed{p^4 - 5p^2 - 5p + 7} \]