1. We are asked to multiply the complex numbers $(3+2i)$ and $(4+5i)$. 2. Use the distributive property (FOIL method): $$ (3+2i)(4+5i) = 3 \times 4 + 3 \times 5i + 2i \times 4 + 2i \times 5i $$ 3. Calculate each term: $$ 3 \times 4 = 12 $$ $$ 3 \times 5i = 15i $$ $$ 2i \times 4 = 8i $$ $$ 2i \times 5i = 10i^2 $$ 4. Recall that $i^2 = -1$, so: $$ 10i^2 = 10 \times (-1) = -10 $$ 5. Sum all terms: $$ 12 + 15i + 8i - 10 = (12 - 10) + (15i + 8i) = 2 + 23i $$ 6. Therefore, the product is $2 + 23i$. The correct answer is option a.