1. **State the problem:** Simplify the expression $$\frac{a}{a^2 + 3a - 18} - \frac{1}{a^2 - 3a}$$. 2. **Factor the denominators:** - Factor $$a^2 + 3a - 18$$: find two numbers that multiply to $$-18$$ and add to $$3$$, which are $$6$$ and $$-3$$. $$a^2 + 3a - 18 = (a + 6)(a - 3)$$. - Factor $$a^2 - 3a$$: $$a^2 - 3a = a(a - 3)$$. 3. **Rewrite the expression with factored denominators:** $$\frac{a}{(a + 6)(a - 3)} - \frac{1}{a(a - 3)}$$. 4. **Find the common denominator:** The least common denominator (LCD) is $$a(a + 6)(a - 3)$$. 5. **Rewrite each fraction with the LCD:** $$\frac{a \cdot a}{a(a + 6)(a - 3)} - \frac{1 \cdot (a + 6)}{a(a + 6)(a - 3)} = \frac{a^2}{a(a + 6)(a - 3)} - \frac{a + 6}{a(a + 6)(a - 3)}$$. 6. **Combine the fractions:** $$\frac{a^2 - (a + 6)}{a(a + 6)(a - 3)} = \frac{a^2 - a - 6}{a(a + 6)(a - 3)}$$. 7. **Factor the numerator:** $$a^2 - a - 6$$ factors as $$(a - 3)(a + 2)$$ because $$-3 \times 2 = -6$$ and $$-3 + 2 = -1$$. 8. **Rewrite the expression:** $$\frac{(a - 3)(a + 2)}{a(a + 6)(a - 3)}$$. 9. **Cancel common factors:** Cancel $$(a - 3)$$ from numerator and denominator: $$\frac{\cancel{(a - 3)}(a + 2)}{a(a + 6)\cancel{(a - 3)}} = \frac{a + 2}{a(a + 6)}$$. **Final answer:** $$\boxed{\frac{a + 2}{a(a + 6)}}$$