Factor. \[ d^{2}-2 d-3 \]

To factor the quadratic expression \( d^{2} - 2d - 3 \), follow these steps: 1. **Identify the Coefficients:** The quadratic is in the form \( ax^{2} + bx + c \), where: - \( a = 1 \) (coefficient of \( d^2 \)) - \( b = -2 \) (coefficient of \( d \)) - \( c = -3 \) (constant term) 2. **Find Two Numbers That Multiply to \( a \times c = 1 \times (-3) = -3 \) and Add to \( b = -2 \):** We need two numbers, let's call them \( m \) and \( n \), such that: \[ m \times n = -3 \quad \text{and} \quad m + n = -2 \] The numbers \( 1 \) and \( -3 \) satisfy these conditions because: \[ 1 \times (-3) = -3 \quad \text{and} \quad 1 + (-3) = -2 \] 3. **Rewrite the Middle Term Using \( m \) and \( n \):** Replace the middle term \( -2d \) with \( 1d - 3d \): \[ d^{2} + 1d - 3d - 3 \] 4. **Factor by Grouping:** - Group the first two terms and the last two terms: \[ (d^{2} + 1d) + (-3d - 3) \] - Factor out the common factors from each group: \[ d(d + 1) - 3(d + 1) \] - Notice that \( (d + 1) \) is a common factor: \[ (d + 1)(d - 3) \] 5. **Final Factored Form:** \[ d^{2} - 2d - 3 = (d + 1)(d - 3) \] **Verification:** To ensure the factoring is correct, expand the factored form: \[ (d + 1)(d - 3) = d \cdot d + d \cdot (-3) + 1 \cdot d + 1 \cdot (-3) = d^{2} - 3d + d - 3 = d^{2} - 2d - 3 \] The original quadratic is successfully factored as: \[ \boxed{(d + 1)(d - 3)} \]