1. **State the problem:** Factorise fully the expression $$45x^2y - 5yz^2$$. 2. **Identify common factors:** Both terms have a common factor of $$5y$$. 3. **Factor out the common factor:** $$45x^2y - 5yz^2 = 5y(\cancel{9x^2} - \cancel{1z^2})$$ 4. **Simplify inside the parentheses:** $$5y(9x^2 - z^2)$$ 5. **Recognize difference of squares:** $$9x^2 - z^2 = (3x)^2 - (z)^2 = (3x - z)(3x + z)$$ 6. **Write the fully factorised form:** $$5y(3x - z)(3x + z)$$ **Final answer:** $$5y(3x - z)(3x + z)$$