1. Problem: Multiply the monomial by the polynomial for the first expression: $a(4a + 3)$. 2. Formula: Use the distributive property $a(b + c) = ab + ac$ to multiply each term inside the parentheses by $a$. 3. Work: $$a(4a + 3) = a \cdot 4a + a \cdot 3 = 4a^2 + 3a$$ 4. Explanation: Multiply the coefficients and add exponents for like bases. Here, $a \cdot 4a = 4a^2$ because $a^1 \times a^1 = a^{1+1} = a^2$. 5. Final answer: $$\boxed{4a^2 + 3a}$$