Divide and simplify. \[ \frac{9 b+18}{2 a+6} \div \frac{a b-3 b+2 a-6}{a^{2}-6 a+9} \]

Factor the expression by following steps: - step0: Factor: \(2a+6\) - step1: Factor the expression: \(2\left(a+3\right)\) Factor the expression \( a^{2}-6 a+9 \). Factor the expression by following steps: - step0: Factor: \(a^{2}-6a+9\) - step1: Factor the expression: \(\left(a-3\right)^{2}\) Factor the expression \( a b-3 b+2 a-6 \). Factor the expression by following steps: - step0: Factor: \(ab-3b+2a-6\) - step1: Calculate: \(ab+2a-3b-6\) - step2: Rewrite the expression: \(ab+a\times 2-3b-3\times 2\) - step3: Factor the expression: \(a\left(b+2\right)-3\left(b+2\right)\) - step4: Factor the expression: \(\left(a-3\right)\left(b+2\right)\) Factor the expression \( 9 b+18 \). Factor the expression by following steps: - step0: Factor: \(9b+18\) - step1: Factor the expression: \(9\left(b+2\right)\) Calculate or simplify the expression \( \frac{9(b+2)}{2(a+3)} \div \frac{(a-3)(b+2)}{(a-3)^{2}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{9\left(b+2\right)}{2\left(a+3\right)}\div \frac{\left(a-3\right)\left(b+2\right)}{\left(a-3\right)^{2}}\) - step1: Multiply by \(a^{-n}:\) \(\frac{9\left(b+2\right)}{2\left(a+3\right)}\div \left(\left(a-3\right)\left(b+2\right)\left(a-3\right)^{-2}\right)\) - step2: Multiply: \(\frac{9\left(b+2\right)}{2\left(a+3\right)}\div \left(\left(a-3\right)^{-1}\left(b+2\right)\right)\) - step3: Rewrite the expression: \(\frac{9\left(b+2\right)}{2\left(a+3\right)}\div \frac{b+2}{a-3}\) - step4: Multiply by the reciprocal: \(\frac{9\left(b+2\right)}{2\left(a+3\right)}\times \frac{a-3}{b+2}\) - step5: Reduce the fraction: \(\frac{9}{2\left(a+3\right)}\times \left(a-3\right)\) - step6: Multiply the terms: \(\frac{9\left(a-3\right)}{2\left(a+3\right)}\) - step7: Simplify: \(\frac{9a-27}{2\left(a+3\right)}\) - step8: Simplify: \(\frac{9a-27}{2a+6}\) To simplify the expression \[ \frac{9 b+18}{2 a+6} \div \frac{a b-3 b+2 a-6}{a^{2}-6 a+9}, \] we will follow these steps: 1. **Factor the expressions**: - The numerator \(9b + 18\) can be factored as \(9(b + 2)\). - The denominator \(2a + 6\) can be factored as \(2(a + 3)\). - The numerator of the second fraction \(ab - 3b + 2a - 6\) can be factored as \((a - 3)(b + 2)\). - The denominator \(a^2 - 6a + 9\) can be factored as \((a - 3)^2\). 2. **Rewrite the division as multiplication**: \[ \frac{9(b + 2)}{2(a + 3)} \div \frac{(a - 3)(b + 2)}{(a - 3)^2} = \frac{9(b + 2)}{2(a + 3)} \times \frac{(a - 3)^2}{(a - 3)(b + 2)}. \] 3. **Cancel common factors**: - The \(b + 2\) in the numerator and denominator cancels out. - One \(a - 3\) from the numerator cancels with one \(a - 3\) from the denominator. 4. **Resulting expression**: After canceling, we have: \[ \frac{9(a - 3)}{2(a + 3)}. \] 5. **Final simplification**: The final simplified expression is: \[ \frac{9a - 27}{2a + 6}. \] Thus, the simplified result is \[ \frac{9a - 27}{2a + 6}. \]