1. **State the problem:** We need to simplify and solve the given complex fractional expressions involving polynomials in $s$ and $t$. 2. **Identify the expressions:** The problem shows multiple fractions such as: - $\frac{d}{5 + 2} = (2s + 1)(5 + 2)(5 + 1)$ - $\frac{5}{(2s + 1)(5 + 2)}$ - $\frac{5}{2(s + 2)(5 + 2)}$ - $\frac{5}{2(s + 2)}$ - $\frac{3}{s^2 + 3s + 2}$ - $\frac{3}{s^2 + 2st + s + 2}$ - $\frac{3}{s(s + 2t) + \frac{1}{s + 2}}$ - $\frac{1}{(s + 2)(s + 1)}$ - $\frac{1}{3}$ - $\frac{(5 + 2)(s + 1)}{+ 5}$ - $\frac{5}{(s + 1)(s - 2)}$ - $\frac{+ 5}{s - 5 - 2}$ - $s^2 - 2st + s - 2 + 8$ over $3$ 3. **Simplify each expression step-by-step:** - Simplify constants: $5 + 2 = 7$, $5 + 1 = 6$ - For $\frac{d}{5 + 2} = (2s + 1)(5 + 2)(5 + 1)$, rewrite as: $$\frac{d}{7} = (2s + 1) \times 7 \times 6 = 42(2s + 1)$$ Multiply both sides by 7: $$d = 7 \times 42 (2s + 1) = 294 (2s + 1)$$ - For $\frac{5}{(2s + 1)(5 + 2)} = \frac{5}{(2s + 1) \times 7} = \frac{5}{7(2s + 1)}$ - For $\frac{5}{2(s + 2)(5 + 2)} = \frac{5}{2 (s + 2) \times 7} = \frac{5}{14 (s + 2)}$ - For $\frac{5}{2(s + 2)}$ remains as is. - Factor $s^2 + 3s + 2$: $$s^2 + 3s + 2 = (s + 1)(s + 2)$$ So $\frac{3}{s^2 + 3s + 2} = \frac{3}{(s + 1)(s + 2)}$ - For $s^2 + 2st + s + 2$, group terms: $$s^2 + 2st + s + 2 = s(s + 2t) + (s + 2)$$ Factor $s + 2$: $$= (s + 2)(s + 1)$$ So $\frac{3}{s^2 + 2st + s + 2} = \frac{3}{(s + 2)(s + 1)}$ - For $\frac{3}{s(s + 2t) + \frac{1}{s + 2}}$, rewrite denominator as: $$s(s + 2t) + \frac{1}{s + 2} = \frac{s(s + 2t)(s + 2) + 1}{s + 2}$$ So the fraction becomes: $$\frac{3}{\frac{s(s + 2t)(s + 2) + 1}{s + 2}} = 3 \times \frac{s + 2}{s(s + 2t)(s + 2) + 1}$$ - For $\frac{1}{(s + 2)(s + 1)}$ remains as is. - For $\frac{1}{3}$ remains as is. - For $\frac{(5 + 2)(s + 1)}{+ 5}$, interpret as: $$\frac{7(s + 1)}{5}$$ - For $\frac{5}{(s + 1)(s - 2)}$ remains as is. - For $\frac{+ 5}{s - 5 - 2} = \frac{5}{s - 7}$ - For $\frac{s^2 - 2st + s - 2 + 8}{3} = \frac{s^2 - 2st + s + 6}{3}$ 4. **Summary of simplified expressions:** - $d = 294 (2s + 1)$ - $\frac{5}{7(2s + 1)}$ - $\frac{5}{14 (s + 2)}$ - $\frac{5}{2(s + 2)}$ - $\frac{3}{(s + 1)(s + 2)}$ - $\frac{3}{(s + 2)(s + 1)}$ - $3 \times \frac{s + 2}{s(s + 2t)(s + 2) + 1}$ - $\frac{1}{(s + 2)(s + 1)}$ - $\frac{1}{3}$ - $\frac{7(s + 1)}{5}$ - $\frac{5}{(s + 1)(s - 2)}$ - $\frac{5}{s - 7}$ - $\frac{s^2 - 2st + s + 6}{3}$ 5. **Explanation:** We factored quadratics, combined like terms, and simplified denominators to express all fractions in simplest polynomial factored form. **Final answer:** The expressions are simplified as above.