1. **Problem:** Factorise completely: $3x^2 - 2xy - 8y^2$. 2. **Formula and rules:** To factorise a quadratic expression $ax^2 + bx + c$, we look for two numbers that multiply to $a \times c$ and add to $b$. 3. **Step 1:** Identify coefficients: $a=3$, $b=-2$, $c=-8$. 4. **Step 2:** Calculate $a \times c = 3 \times (-8) = -24$. 5. **Step 3:** Find two numbers that multiply to $-24$ and add to $-2$: these are $4$ and $-6$ because $4 \times (-6) = -24$ and $4 + (-6) = -2$. 6. **Step 4:** Rewrite the middle term using these numbers: $$3x^2 + 4xy - 6xy - 8y^2$$ 7. **Step 5:** Group terms: $$ (3x^2 + 4xy) + (-6xy - 8y^2) $$ 8. **Step 6:** Factor each group: $$ x(3x + 4y) - 2y(3x + 4y) $$ 9. **Step 7:** Factor out the common binomial: $$ (3x + 4y)(x - 2y) $$ **Final answer:** $$\boxed{(3x + 4y)(x - 2y)}$$