b) \( x^{2}+9 x-36 \)

To factor the quadratic expression \( x^{2} + 9x - 36 \), follow these steps: ### Step 1: Identify the Coefficients The quadratic equation is in the form \( ax^{2} + bx + c \), where: - \( a = 1 \) - \( b = 9 \) - \( c = -36 \) ### Step 2: Find Two Numbers that Multiply to \( a \times c \) and Add to \( b \) We need to find two numbers that: - Multiply to \( 1 \times (-36) = -36 \) - Add up to \( 9 \) **Possible Pairs:** - \( 12 \) and \( -3 \) (since \( 12 \times (-3) = -36 \) and \( 12 + (-3) = 9 \)) ### Step 3: Rewrite the Middle Term Using These Numbers Express the quadratic as: \[ x^{2} + 12x - 3x - 36 \] ### Step 4: Factor by Grouping Group the terms: \[ (x^{2} + 12x) + (-3x - 36) \] Factor out the greatest common factor from each group: \[ x(x + 12) - 3(x + 12) \] ### Step 5: Factor Out the Common Binomial \[ (x - 3)(x + 12) \] ### Final Factored Form \[ x^{2} + 9x - 36 = (x + 12)(x - 3) \] ### Verification (Optional) To ensure the factoring is correct, expand the factors: \[ (x + 12)(x - 3) = x(x) + x(-3) + 12(x) + 12(-3) = x^{2} - 3x + 12x - 36 = x^{2} + 9x - 36 \] Which matches the original expression. **Therefore, the factored form is:** \[ (x + 12)(x - 3) \]