1. **State the problem:** Simplify or factor the quadratic expression $4y^2 - 3y - 27$. 2. **Formula and rules:** To factor a quadratic expression $ay^2 + by + c$, we look for two numbers that multiply to $a \times c$ and add to $b$. 3. **Calculate product and sum:** Here, $a=4$, $b=-3$, and $c=-27$. So, $a \times c = 4 \times (-27) = -108$. 4. We need two numbers that multiply to $-108$ and add to $-3$. 5. The pair is $9$ and $-12$ because $9 \times (-12) = -108$ and $9 + (-12) = -3$. 6. **Rewrite the middle term:** $$4y^2 + 9y - 12y - 27$$ 7. **Group terms:** $$(4y^2 + 9y) + (-12y - 27)$$ 8. **Factor each group:** $$y(4y + 9) - 3(4y + 9)$$ 9. **Factor out common binomial:** $$(4y + 9)(y - 3)$$ 10. **Final answer:** The factored form of $4y^2 - 3y - 27$ is $$\boxed{(4y + 9)(y - 3)}$$.