Factor completely. \( 9 f^{2}+7 f-2 \)

To factor the quadratic expression \( 9f^{2} + 7f - 2 \) completely, we'll look for two numbers that multiply to \( 9 \times -2 = -18 \) and add to \( 7 \). The numbers \( 9 \) and \( -2 \) fit these criteria. We can split the middle term: \( 9f^{2} + 9f - 2f - 2 \). Now, group the terms: \( (9f^{2} + 9f) + (-2f - 2) \). Factor out the common terms in each group: \( 9f(f + 1) - 2(f + 1) \). Now, notice that \( f + 1 \) is a common factor: \( (9f - 2)(f + 1) \). Thus, the complete factorization of \( 9f^{2} + 7f - 2 \) is \( (9f - 2)(f + 1) \).