1. **State the problem:** We are given the quadratic expression $$2x^2 - 6x - 20$$ and asked to fill in the gaps in the expression $$2x^2 - 6x - 20 = \square (x^2 - \square x - \square)$$ Then, we need to fully factorise $$2x^2 - 6x - 20$$ in the form $$a(x + b)(x + c)$$. 2. **Fill in the gaps:** We start by factoring out the greatest common factor (GCF) from the quadratic expression. The GCF of $$2x^2$$, $$-6x$$, and $$-20$$ is 2. So, we write: $$2x^2 - 6x - 20 = 2(x^2 - 3x - 10)$$ Thus, the filled gaps are: - First gap: 2 - Second gap: 3 - Third gap: 10 3. **Factorise the quadratic inside the parentheses:** We now factorise $$x^2 - 3x - 10$$. We look for two numbers that multiply to $$-10$$ and add to $$-3$$. These numbers are $$-5$$ and $$2$$ because: $$-5 \times 2 = -10$$ $$-5 + 2 = -3$$ So, $$x^2 - 3x - 10 = (x - 5)(x + 2)$$ 4. **Write the full factorisation:** Substitute back: $$2(x^2 - 3x - 10) = 2(x - 5)(x + 2)$$ 5. **Final answer:** The fully factorised form is: $$2(x - 5)(x + 2)$$ --- **Summary:** - Filled gaps: $$2, 3, 10$$ - Factorised form: $$2(x - 5)(x + 2)$$