To solve the factorization $x^2 + 4x + 3 = (x + 1)(x + \_\_\_)$, we need to determine the missing number in the second factor.

Step-by-Step Solution:

1. Expand the expression on the right-hand side: $(x + 1)(x + b) = x^2 + (1 + b)x + b$ Here, $b$ is the missing number.

2. Compare coefficients with $x^2 + 4x + 3$:

   • From $x^2 + 4x + 3$, the coefficient of $x$ is $4$, and the constant term is $3$.
   • For the coefficient of $x$: $1 + b = 4$. Solve for $b$: $b = 4 - 1 = 3$

3. Verify the constant term: The product of the constants in the factors, $1 \times b$, must equal $3$. Substituting $b = 3$, we confirm: $1 \times 3 = 3$

Final Answer:

The missing number is 3.

Complete Factorization:

$x^2 + 4x + 3 = (x + 1)(x + 3)$

Would you like further clarification or details about factorization?

Here are 5 related questions for practice:

1. How do you factorize $x^2 + 5x + 6$?
2. How can you check if your factorization is correct?
3. Can every quadratic expression be factorized into two linear factors?
4. What is the significance of the constant term in quadratic factorization?
5. How do you factorize quadratics when the leading coefficient is not 1?

Tip: Always verify your factorization by expanding the factors back to their original expression!