1. Stating the problem: Simplify the expression $ (3)(4) + \left((4)(5) - (1)(2)\right) \frac{3}{8} $. 2. Calculate the products inside the parentheses: $$ (3)(4) = 12 $$ $$ (4)(5) = 20 $$ $$ (1)(2) = 2 $$ 3. Substitute these values back into the expression: $$ 12 + (20 - 2) \frac{3}{8} $$ 4. Simplify inside the parentheses: $$ 12 + 18 \frac{3}{8} $$ 5. Multiply 18 by $\frac{3}{8}$: $$ 18 \times \frac{3}{8} = \frac{18 \times 3}{8} = \frac{54}{8} $$ 6. Simplify the fraction $\frac{54}{8}$ by dividing numerator and denominator by 2: $$ \frac{\cancel{54}^{27}}{\cancel{8}^{4}} $$ 7. Now the expression is: $$ 12 + \frac{27}{4} $$ 8. Convert 12 to a fraction with denominator 4: $$ \frac{12 \times 4}{4} = \frac{48}{4} $$ 9. Add the fractions: $$ \frac{48}{4} + \frac{27}{4} = \frac{48 + 27}{4} = \frac{75}{4} $$ 10. Final answer: $$ \frac{75}{4} $$ or as a mixed number $18 \frac{3}{4}$.