1. **Problem:** Factor the quadratic expression $3x^2 - 11x - 4$. 2. **Formula and rules:** To factor a quadratic $ax^2 + bx + c$, find two numbers that multiply to $a \times c$ and add to $b$. 3. For $3x^2 - 11x - 4$, $a=3$, $b=-11$, $c=-4$. Calculate $a \times c = 3 \times (-4) = -12$. 4. Find two numbers that multiply to $-12$ and add to $-11$: these are $-12$ and $1$. 5. Rewrite the middle term: $3x^2 - 12x + x - 4$. 6. Group terms: $(3x^2 - 12x) + (x - 4)$. 7. Factor each group: $3x(x - 4) + 1(x - 4)$. 8. Factor out common binomial: $(3x + 1)(x - 4)$. **Final answer:** $3x^2 - 11x - 4 = (3x + 1)(x - 4)$.