To solve the problem, let's break it down into two parts as specified.

### Part a: Calculate the partial derivatives of \( M \)

The expression for \( M \) is given by:

\[
M = 3x^3 + 4x^2y - 5xy^2 + 7xy - y^3
\]

1. **Partial derivative with respect to \( x \) (\( \frac{\partial M}{\partial x} \))**:

- Differentiate \( M \) with respect to \( x \):

\[
\frac{\partial M}{\partial x} = 9x^2 + 8xy - 5y^2 + 7y
\]

2. **Partial derivative with respect to \( y \) (\( \frac{\partial M}{\partial y} \))**:

- Differentiate \( M \) with respect to \( y \):

\[
\frac{\partial M}{\partial y} = 4x^2 - 10xy + 7x - 3y^2
\]

### Part b: Evaluating the derivatives at \( x = -2, y = -3 \)

1. **Evaluate \( \frac{\partial M}{\partial x} \) at \( x = -2 \) and \( y = -3 \)**:

\[
\frac{\partial M}{\partial x}(-2, -3) = 9(-2)^2 + 8(-2)(-3) - 5(-3)^2 + 7(-3)
\]
\[
= 9(4) + 48 - 5(9) - 21
\]
\[
= 36 + 48 - 45 - 21
\]
\[
= 18
\]

2. **Evaluate \( \frac{\partial M}{\partial y} \) at \( x = -2 \) and \( y = -3 \)**:

\[
\frac{\partial M}{\partial y}(-2, -3) = 4(-2)^2 - 10(-2)(-3) + 7(-2) - 3(-3)^2
\]
\[
= 4(4) - 60 - 14 - 27
\]
\[
= 16 - 60 - 14 - 27
\]
\[
= -85
\]

### Summary of Results:
- \( \frac{\partial M}{\partial x}(-2, -3) = 18 \)
- \( \frac{\partial M}{\partial y}(-2, -3) = -85 \)

This completes the calculation of the partial derivatives and their evaluations at the specified points.