Factor the trinomial \( 42 x^{2}-85 x+42 \)

To factor the trinomial \( 42x^{2} - 85x + 42 \), we can follow these steps: 1. **Identify Coefficients:** - \( a = 42 \) - \( b = -85 \) - \( c = 42 \) 2. **Multiply \( a \) and \( c \):** - \( a \times c = 42 \times 42 = 1764 \) 3. **Find Two Numbers that Multiply to 1764 and Add to -85:** - After searching for factor pairs of 1764, we find: - **-36 and -49** because: - \( -36 \times -49 = 1764 \) - \( -36 + (-49) = -85 \) 4. **Rewrite the Middle Term Using These Numbers:** \[ 42x^{2} - 36x - 49x + 42 \] 5. **Factor by Grouping:** \[ (42x^{2} - 36x) + (-49x + 42) \] \[ 6x(7x - 6) - 7(7x - 6) \] 6. **Factor Out the Common Binomial:** \[ (6x - 7)(7x - 6) \] **Final Factored Form:** \[ 42x^{2} - 85x + 42 = (6x - 7)(7x - 6) \] Answer: \(42x^{2} - 85x + 42 = (6x - 7)\, (7x - 6)\)