1. **Problem:** Factorize the expression $9x^2 + 8x - 1$. 2. **Formula and rules:** To factorize a quadratic expression $ax^2 + bx + c$, we look for two numbers that multiply to $ac$ and add to $b$. 3. **Intermediate work:** Here, $a=9$, $b=8$, and $c=-1$. We need two numbers that multiply to $9 \times (-1) = -9$ and add to $8$. 4. The numbers are $9$ and $-1$ because $9 \times (-1) = -9$ and $9 + (-1) = 8$. 5. Rewrite the middle term using these numbers: $$9x^2 + 9x - x - 1$$ 6. Group terms: $$ (9x^2 + 9x) - (x + 1) $$ 7. Factor each group: $$ 9x(x + 1) - 1(x + 1) $$ 8. Factor out the common binomial: $$ (9x - 1)(x + 1) $$ 9. **Answer:** The factorization of $9x^2 + 8x - 1$ is $$\boxed{(9x - 1)(x + 1)}$$.