1. **State the problem:** Factor the quadratic expression $3y^2 + 4y - 7$. 2. **Recall the factoring formula:** For a quadratic $ay^2 + by + c$, we look for two numbers that multiply to $a \times c$ and add to $b$. 3. Here, $a=3$, $b=4$, and $c=-7$. Calculate $a \times c = 3 \times (-7) = -21$. 4. Find two numbers that multiply to $-21$ and add to $4$. These numbers are $7$ and $-3$ because $7 \times (-3) = -21$ and $7 + (-3) = 4$. 5. Rewrite the middle term using these numbers: $$3y^2 + 7y - 3y - 7$$ 6. Group terms: $$ (3y^2 + 7y) - (3y + 7) $$ 7. Factor each group: $$ y(3y + 7) - 1(3y + 7) $$ 8. Factor out the common binomial: $$ (3y + 7)(y - 1) $$ **Final answer:** The factored form of $3y^2 + 4y - 7$ is $$ (3y + 7)(y - 1) $$.