8) \( 2 n^{2}+n-21 \)

To factor the quadratic expression \( 2n^{2} + n - 21 \), we can follow these steps: 1. **Identify the coefficients:** - \( a = 2 \) (coefficient of \( n^2 \)) - \( b = 1 \) (coefficient of \( n \)) - \( c = -21 \) (constant term) 2. **Find two numbers that multiply to \( a \times c = 2 \times (-21) = -42 \) and add up to \( b = 1 \):** After checking possible pairs, we find: - \( -6 \times 7 = -42 \) - \( -6 + 7 = 1 \) 3. **Rewrite the middle term using these numbers:** \[ 2n^{2} + n - 21 = 2n^{2} - 6n + 7n - 21 \] 4. **Factor by grouping:** \[ = (2n^{2} - 6n) + (7n - 21) \] \[ = 2n(n - 3) + 7(n - 3) \] 5. **Factor out the common binomial factor \((n - 3)\):** \[ = (2n + 7)(n - 3) \] **Final Factored Form:** \[ 2n^{2} + n - 21 = (2n + 7)(n - 3) \]