\begin{tabular}{l} 5. \( \frac{x^{2}+12 x+20}{9 x+90} \cdot \frac{9}{x+3} \) \\ 6. \( \frac{1}{x+2} \div \frac{x-8}{x^{2}-14 x+48} \) \\ \hline\end{tabular}

We’ll simplify each expression step by step. ───────────────────────────── Problem 5. Simplify   ( (x² + 12x + 20) / (9x + 90) ) · (9 / (x + 3) ) 1. Factor the polynomial in the numerator and factor out common factors from the denominator:  • x² + 12x + 20 factors as (x + 2)(x + 10)  • 9x + 90 factors as 9(x + 10) 2. Substitute the factorizations into the expression:   [(x + 2)(x + 10)] / [9(x + 10)] · [9 / (x + 3)] 3. Cancel the common factor (x + 10) (with the condition x ≠ –10) and cancel the 9:   = (x + 2) / (x + 3) So, the simplified expression is (x + 2)/(x + 3) with the additional restrictions that x ≠ –10 (from 9x + 90) and x ≠ –3 (from the factor x + 3). ───────────────────────────── Problem 6. Simplify   (1 / (x + 2)) ÷ ((x – 8) / (x² – 14x + 48)) 1. Rewrite the division as multiplication by the reciprocal:   = (1 / (x + 2)) · ((x² – 14x + 48) / (x – 8)) 2. Factor the quadratic in the numerator:  • x² – 14x + 48 factors as (x – 6)(x – 8) 3. Substitute in the factors:   = (1 / (x + 2)) · ( (x – 6)(x – 8) / (x – 8) ) 4. Cancel the common factor (x – 8) (with the condition x ≠ 8):   = (x – 6) / (x + 2) Thus, the simplified expression is (x – 6)/(x + 2) with the restrictions that x ≠ –2 (from the factor x + 2), x ≠ 8 (canceled) and note that the factored quadratic also implies x ≠ 6 (from (x – 6) if it had appeared in a denominator originally). ───────────────────────────── Final Answers: 5. (x + 2)/(x + 3), with x ≠ –10 and x ≠ –3. 6. (x – 6)/(x + 2), with x ≠ –2 and x ≠ 8. (Always check that any cancellation is done under the condition that the canceled factors are not zero.)