1. **State the problem:** Simplify the expression $$\frac{(x+3)(x-2)}{(x+3)(x-1)} - \frac{7-2}{x-1}$$ 2. **Rewrite the expression:** The second fraction numerator simplifies to $7-2=5$, so the expression becomes: $$\frac{(x+3)(x-2)}{(x+3)(x-1)} - \frac{5}{x-1}$$ 3. **Find a common denominator:** The denominators are $(x+3)(x-1)$ and $x-1$. The least common denominator (LCD) is $(x+3)(x-1)$. 4. **Rewrite the second fraction with the LCD:** Multiply numerator and denominator by $(x+3)$: $$\frac{5}{x-1} = \frac{5(x+3)}{(x-1)(x+3)}$$ 5. **Rewrite the expression with common denominator:** $$\frac{(x+3)(x-2)}{(x+3)(x-1)} - \frac{5(x+3)}{(x+3)(x-1)} = \frac{(x+3)(x-2) - 5(x+3)}{(x+3)(x-1)}$$ 6. **Factor out $(x+3)$ in the numerator:** $$\frac{(x+3)((x-2) - 5)}{(x+3)(x-1)}$$ 7. **Simplify inside the parentheses:** $$(x-2) - 5 = x - 7$$ 8. **Expression becomes:** $$\frac{(x+3)(x-7)}{(x+3)(x-1)}$$ 9. **Cancel common factor $(x+3)$:** $$\frac{\cancel{(x+3)}(x-7)}{\cancel{(x+3)}(x-1)} = \frac{x-7}{x-1}$$ **Final answer:** $$\boxed{\frac{x-7}{x-1}}$$