1. **Problem statement:** (a) Factorise $x^2 - 16$ (b) Factorise $3x^2 + 7x - 20$ (c) Simplify $\frac{x^2 - 16}{3x^2 + 7x - 20}$ --- 2. **Formula and rules:** - Difference of squares: $a^2 - b^2 = (a - b)(a + b)$ - Factorising quadratic $ax^2 + bx + c$ by splitting the middle term or using the AC method. - Simplifying rational expressions by cancelling common factors. --- 3. **Step-by-step solution:** **(a) Factorise $x^2 - 16$:** - Recognize $x^2 - 16$ as a difference of squares: $x^2 - 4^2$ - Apply the formula: $x^2 - 4^2 = (x - 4)(x + 4)$ **(b) Factorise $3x^2 + 7x - 20$:** - Multiply $a$ and $c$: $3 \times (-20) = -60$ - Find two numbers that multiply to $-60$ and add to $7$: $12$ and $-5$ - Split the middle term: $3x^2 + 12x - 5x - 20$ - Group terms: $(3x^2 + 12x) + (-5x - 20)$ - Factor each group: $3x(x + 4) - 5(x + 4)$ - Factor out common binomial: $(x + 4)(3x - 5)$ **(c) Simplify $\frac{x^2 - 16}{3x^2 + 7x - 20}$:** - Substitute the factored forms: $$\frac{(x - 4)(x + 4)}{(x + 4)(3x - 5)}$$ - Cancel the common factor $(x + 4)$ (assuming $x \neq -4$ to avoid division by zero): $$\frac{x - 4}{3x - 5}$$ --- 4. **Final answers:** (a) $(x - 4)(x + 4)$ (b) $(x + 4)(3x - 5)$ (c) $\frac{x - 4}{3x - 5}$ This completes the factorisation and simplification.