To solve the problem given \( x + y = -5 \) and \( xy = 7 \), we can use the relationships between the sums and products of roots of polynomials.

### Part (a): Calculate \( x^2 + y^2 \)

Using the identity:
\[
x^2 + y^2 = (x + y)^2 - 2xy
\]
Substitute the given values:
\[
x^2 + y^2 = (-5)^2 - 2 \cdot 7
\]
\[
= 25 - 14 = 11
\]

### Part (b): Calculate \( x^3 + y^3 \)

Using the identity:
\[
x^3 + y^3 = (x + y)(x^2 - xy + y^2)
\]
First, calculate \( x^2 - xy + y^2 \):
\[
x^2 - xy + y^2 = x^2 + y^2 - xy = 11 - 7 = 4
\]
Now substituting into the formula for \( x^3 + y^3 \):
\[
x^3 + y^3 = (x + y)(x^2 - xy + y^2) = (-5)(4) = -20
\]

### Summary of Results:
- \( x^2 + y^2 = 11 \)
- \( x^3 + y^3 = -20 \)