1. The problem involves finding the product of the binomials $(4x + 1)(x + 1)$ and simplifying the expression. 2. The formula for multiplying two binomials is given by the distributive property: $$(a + b)(c + d) = ac + ad + bc + bd$$ 3. Applying this to $(4x + 1)(x + 1)$, we multiply each term in the first binomial by each term in the second binomial: $$4x \cdot x + 4x \cdot 1 + 1 \cdot x + 1 \cdot 1$$ 4. Simplify each term: $$4x^2 + 4x + x + 1$$ 5. Combine like terms: $$4x^2 + (4x + x) + 1 = 4x^2 + 5x + 1$$ 6. The expression simplifies to: $$\boxed{4x^2 + 5x + 1}$$ This matches the expression shown in the problem and confirms the factorization and multiplication steps.