1. **State the problem:** Factorise the quadratic expression $3x^2 + 5x - 12$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multiply to $a \times c$ and add to $b$. 3. **Calculate the product and sum:** Here, $a = 3$, $b = 5$, and $c = -12$. So, $a \times c = 3 \times (-12) = -36$. 4. **Find two numbers:** We need two numbers that multiply to $-36$ and add to $5$. These numbers are $9$ and $-4$ because $9 \times (-4) = -36$ and $9 + (-4) = 5$. 5. **Rewrite the middle term:** Rewrite $5x$ as $9x - 4x$: $$3x^2 + 9x - 4x - 12$$ 6. **Group terms:** $$(3x^2 + 9x) + (-4x - 12)$$ 7. **Factor each group:** $$3x(x + 3) - 4(x + 3)$$ 8. **Factor out the common binomial:** $$(3x - 4)(x + 3)$$ **Final answer:** The factorised form of $3x^2 + 5x - 12$ is $$\boxed{(3x - 4)(x + 3)}$$.