1. **Problem:** Factor the polynomial $3x^2 + 14x - 5$ completely. 2. **Formula and rules:** To factor a quadratic $ax^2 + bx + c$, find two numbers that multiply to $a \times c$ and add to $b$. Then split the middle term and factor by grouping. 3. **Intermediate work:** - Here, $a=3$, $b=14$, $c=-5$. - Calculate $a \times c = 3 \times (-5) = -15$. - Find two numbers that multiply to $-15$ and add to $14$: these are $15$ and $-1$. - Rewrite the middle term: $3x^2 + 15x - x - 5$. - Group terms: $(3x^2 + 15x) - (x + 5)$. - Factor each group: $3x(x + 5) - 1(x + 5)$. - Factor out common binomial: $(3x - 1)(x + 5)$. 4. **Final answer:** $$3x^2 + 14x - 5 = (3x - 1)(x + 5)$$