Add the polynomial expressions 5z - 2z^2 - 4z + 5 and 6z^2 + 3 - 7z.

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

1. Write the first polynomial:
\[
5z - 2z^2 - 4z + 5 = -2z^2 + (5z - 4z) + 5 = -2z^2 + 1z + 5
\]

2. Write the second polynomial:
\[
6z^2 + 3 - 7z = 6z^2 + (-7z) + 3
\]

Now, we combine both polynomials:
\[
(-2z^2 + z + 5) + (6z^2 - 7z + 3)
\]

Now, add the like terms:
- Combine the \(z^2\) terms: \(-2z^2 + 6z^2 = 4z^2\)
- Combine the \(z\) terms: \(z - 7z = -6z\)
- Combine the constant terms: \(5 + 3 = 8\)

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

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