1. **State the problem:** Factorise fully the quadratic expression $$6y^2 - 5y - 4$$. 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. **Calculate product and sum:** Here, $$a=6$$, $$b=-5$$, and $$c=-4$$. Calculate product: $$6 \times (-4) = -24$$. We need two numbers that multiply to $$-24$$ and add to $$-5$$. 4. **Find the pair:** The numbers are $$3$$ and $$-8$$ because $$3 \times (-8) = -24$$ and $$3 + (-8) = -5$$. 5. **Rewrite the middle term:** $$6y^2 - 5y - 4 = 6y^2 + 3y - 8y - 4$$ 6. **Group terms:** $$(6y^2 + 3y) + (-8y - 4)$$ 7. **Factor each group:** $$3y(2y + 1) - 4(2y + 1)$$ 8. **Factor out the common binomial:** $$(3y - 4)(2y + 1)$$ **Final answer:** $$6y^2 - 5y - 4 = (3y - 4)(2y + 1)$$.