Factor the trinomial completely. \[ x^{2}+x-72 \]

To factor the trinomial \( x^{2} + x - 72 \) completely, follow these steps: 1. **Identify the coefficients:** - \( a = 1 \) (coefficient of \( x^2 \)) - \( b = 1 \) (coefficient of \( x \)) - \( c = -72 \) (constant term) 2. **Find two numbers that multiply to \( a \times c = 1 \times (-72) = -72 \) and add up to \( b = 1 \).** Let's list the factor pairs of \(-72\): - \( 1 \) and \( -72 \) (sum: \(-71\)) - \( -1 \) and \( 72 \) (sum: \(71\)) - \( 2 \) and \( -36 \) (sum: \(-34\)) - \( -2 \) and \( 36 \) (sum: \(34\)) - \( 3 \) and \( -24 \) (sum: \(-21\)) - \( -3 \) and \( 24 \) (sum: \(21\)) - \( 4 \) and \( -18 \) (sum: \(-14\)) - \( -4 \) and \( 18 \) (sum: \(14\)) - \( 6 \) and \( -12 \) (sum: \(-6\)) - \( -6 \) and \( 12 \) (sum: \(6\)) - \( 8 \) and \( -9 \) (sum: \(-1\)) - \( -8 \) and \( 9 \) (sum: \(1\)) The pair that adds up to \(1\) is \(-8\) and \(9\). 3. **Rewrite the middle term using these numbers:** \[ x^{2} + 9x - 8x - 72 \] 4. **Factor by grouping:** \[ x(x + 9) - 8(x + 9) \] \[ (x - 8)(x + 9) \] **Final Factored Form:** \[ (x - 8)(x + 9) \]