1. **State the problem:** We are given the quadratic expression $$2x^2 - 6x - 20$$ and asked to fill in the gaps in the expression $$2x^2 - 6x - 20 = [ ] (x^2 - [ ] x - [ ])$$. 2. **Identify the common factor:** We look for a common factor in all terms of $$2x^2 - 6x - 20$$. The greatest common factor (GCF) is 2. 3. **Factor out the GCF:** Factoring out 2, we get: $$2x^2 - 6x - 20 = 2(x^2 - 3x - 10)$$ So the gaps are filled as: $$2(x^2 - 3x - 10)$$. 4. **Fully factorise the quadratic inside the parentheses:** We want to factor $$x^2 - 3x - 10$$ into the form $$(x + b)(x + c)$$. 5. **Find two numbers that multiply to $$-10$$ and add to $$-3$$:** These numbers are $$-5$$ and $$2$$ because: $$-5 \times 2 = -10$$ $$-5 + 2 = -3$$ 6. **Write the factorised form:** $$x^2 - 3x - 10 = (x - 5)(x + 2)$$ 7. **Combine with the GCF:** The full factorisation is: $$2(x - 5)(x + 2)$$ **Final answer:** $$2x^2 - 6x - 20 = 2(x - 5)(x + 2)$$