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To add the polynomial expressions \( 3 - 2p - 5p^2 \) and \( p^4 - 3p + 4 \), you need to combine like terms:
1. Write the two polynomials one under the other, aligning like terms:
\[
\begin{array}{r}
3 \\
-2p \\
-5p^2 \\
\hline
\end{array}
+
\begin{array}{r}
p^4 \\
-3p \\
4 \\
\hline
\end{array}
\]
2. Now, combine all the like terms:
- The highest degree term is \( p^4 \).
- The \( p^2 \) term is \( -5p^2 \).
- The \( p \) terms are \( -2p - 3p = -5p \).
- The constant terms are \( 3 + 4 = 7 \).
Putting it all together, we get:
\[
p^4 - 5p^2 - 5p + 7
\]
Thus, the sum of the polynomials \( 3 - 2p - 5p^2 \) and \( p^4 - 3p + 4 \) is:
\[
\boxed{p^4 - 5p^2 - 5p + 7}
\]