b) \( x^{2}+8 x+12 \)

Certainly! Let's work on factoring the quadratic expression: \[ x^{2} + 8x + 12 \] **Step 1: Identify the Coefficients** A quadratic expression is generally in the form: \[ ax^{2} + bx + c \] For the given expression: - \( a = 1 \) - \( b = 8 \) - \( c = 12 \) **Step 2: Find Two Numbers that Multiply to \( a \times c \) and Add to \( b \)** We need two numbers that satisfy: - **Product**: \( a \times c = 1 \times 12 = 12 \) - **Sum**: \( b = 8 \) Let's list the factors of 12: - \( 1 \times 12 \) (Sum: 13) - \( 2 \times 6 \) (Sum: 8) - \( 3 \times 4 \) (Sum: 7) The pair \( 2 \) and \( 6 \) meets both criteria. **Step 3: Write the Factored Form** Using the numbers \( 2 \) and \( 6 \), we can write: \[ x^{2} + 8x + 12 = (x + 2)(x + 6) \] **Verification:** To ensure the factoring is correct, expand the product: \[ (x + 2)(x + 6) = x(x + 6) + 2(x + 6) = x^2 + 6x + 2x + 12 = x^2 + 8x + 12 \] This matches the original expression, confirming that the factorization is correct. **Final Answer:** \[ x^{2} + 8x + 12 = (x + 2)(x + 6) \]