1. The problem asks us to find the number that completes the factorization of the quadratic expression $x^2 + 4x + 3$ in the form $(x + 1)(x + \_\_).$ 2. Start by expanding the right-hand side expression with the unknown number $a$: $$ (x + 1)(x + a) = x^2 + ax + x + a = x^2 + (a + 1)x + a $$ 3. Compare this with the left-hand side expression $x^2 + 4x + 3$. 4. From the coefficients of $x$, we have: $$ a + 1 = 4 $$ 5. Solve for $a$: $$ a = 4 - 1 = 3 $$ 6. Check the constant term: $$ a = 3 $$ which matches the constant term on the left side. 7. Therefore, the number that goes in the gap is $3$. Final answer: $3$