1. Simplify the polynomial expression $a(4a + 3)$. 2. Use the distributive property: $a(b + c) = ab + ac$. 3. Multiply $a$ by each term inside the parentheses: $$a \times 4a = 4a^2$$ $$a \times 3 = 3a$$ 4. Combine the results: $$4a^2 + 3a$$ 5. The expression in standard form is: $$\boxed{4a^2 + 3a}$$