Add the polynomial expressions 5z−2z^2−4z+5

To add the polynomial expressions \( 5z - 2z^2 - 4z + 5 \) and \( 6z^2 + 3 - 7z \), we first need to combine like terms in each expression.

1. **First expression**:
\[
5z - 2z^2 - 4z + 5
\]
Combine like terms:
\[
(-2z^2) + (5z - 4z) + 5 = -2z^2 + z + 5
\]

2. **Second expression**:
\[
6z^2 + 3 - 7z
\]
This expression is already simplified.

Now, we add the two simplified expressions:
\[
(-2z^2 + z + 5) + (6z^2 - 7z + 3)
\]

Now, combine the like terms:
- For \( z^2 \):
\[
-2z^2 + 6z^2 = 4z^2
\]
- For \( z \):
\[
z - 7z = -6z
\]
- For the constant:
\[
5 + 3 = 8
\]

Putting it all together, we get:
\[
4z^2 - 6z + 8
\]

So the sum of the two polynomial expressions is:
\[
\boxed{4z^2 - 6z + 8}
\]