To solve the expression you provided, we need to simplify the following:

\[
\frac{4x}{x+2} + \frac{2-x}{x-3} - \frac{6}{(x-3)(x+2)}
\]

### Steps to Simplify:

1. **Find a Common Denominator**:
The common denominator for all terms is \((x-3)(x+2)\).

2. **Convert Each Fraction**:
- The first term becomes:
\[
\frac{4x(x-3)}{(x+2)(x-3)} = \frac{4x(x-3)}{(x-3)(x+2)}
\]
- The second term becomes:
\[
\frac{(2-x)(x+2)}{(x-3)(x+2)} = \frac{(2-x)(x+2)}{(x-3)(x+2)}
\]
- The third term is already in the proper form:
\[
-\frac{6}{(x-3)(x+2)}
\]

3. **Combine the Fractions**:
\[
\frac{4x(x-3) + (2-x)(x+2) - 6}{(x-3)(x+2)}
\]

4. **Expand the Numerator**:
- For \(4x(x-3)\):
\[
4x^2 - 12x
\]
- For \((2-x)(x+2)\):
\[
2x + 4 - x^2 - 2x = 4 - x^2
\]

Combining:
\[
4x^2 - 12x + 4 - x^2 - 6 = 3x^2 - 12x - 2
\]

5. **Final Expression**:
\[
\frac{3x^2 - 12x - 2}{(x-3)(x+2)}
\]

This is the simplified form of your given expression.