1. **State the problem:** Simplify the expression $$d \left( \frac{3}{8} - \frac{1}{6} m + 5t \right) + \left( \frac{7}{10} m + 9t + \frac{1}{4} \right)$$. 2. **Distribute** $d$ across the terms inside the first parentheses: $$d \times \frac{3}{8} - d \times \frac{1}{6} m + d \times 5t = \frac{3}{8} d - \frac{1}{6} d m + 5 d t$$ 3. **Rewrite the expression:** $$\frac{3}{8} d - \frac{1}{6} d m + 5 d t + \frac{7}{10} m + 9 t + \frac{1}{4}$$ 4. **Group like terms:** - Terms with $m$: $- \frac{1}{6} d m + \frac{7}{10} m$ - Terms with $t$: $5 d t + 9 t$ - Constant terms: $\frac{3}{8} d + \frac{1}{4}$ 5. **Factor $m$ and $t$ where possible:** $$m \left(- \frac{1}{6} d + \frac{7}{10} \right) + t \left(5 d + 9 \right) + \frac{3}{8} d + \frac{1}{4}$$ 6. **Find common denominators to combine coefficients:** - For $m$ coefficients: common denominator is 30 $$- \frac{1}{6} d = - \frac{5}{30} d, \quad \frac{7}{10} = \frac{21}{30}$$ So, $$m \left(- \frac{5}{30} d + \frac{21}{30} \right) = m \frac{21 - 5 d}{30}$$ - For constants, leave as is since they cannot be combined further. 7. **Final simplified expression:** $$\frac{3}{8} d + \frac{1}{4} + m \frac{21 - 5 d}{30} + t (5 d + 9)$$ This is the simplified form with like terms combined and factored where possible.