1. Problem (a): Simplify $$(2^5 - 2)(3^5 + 4^2)$$. 2. Calculate powers: $$2^5 = 32$$ $$3^5 = 243$$ $$4^2 = 16$$ 3. Substitute values: $$(32 - 2)(243 + 16) = 30 \times 259$$ 4. Multiply: $$30 \times 259 = 7770$$ --- 1. Problem (b): Simplify $$(y + 5)(y - 3)$$. 2. Use distributive property (FOIL): $$y \times y = y^2$$ $$y \times (-3) = -3y$$ $$5 \times y = 5y$$ $$5 \times (-3) = -15$$ 3. Combine like terms: $$y^2 - 3y + 5y - 15 = y^2 + 2y - 15$$ --- 1. Problem (c): Simplify $$(3 + 1)^2$$. 2. Calculate inside parentheses: $$3 + 1 = 4$$ 3. Square the result: $$4^2 = 16$$ --- 1. Problem (d): Simplify $$(2 - \sqrt{5})^2$$. 2. Use formula for square of a binomial: $$(a - b)^2 = a^2 - 2ab + b^2$$ 3. Substitute $a=2$, $b=\sqrt{5}$: $$2^2 - 2 \times 2 \times \sqrt{5} + (\sqrt{5})^2 = 4 - 4\sqrt{5} + 5$$ 4. Combine constants: $$4 + 5 - 4\sqrt{5} = 9 - 4\sqrt{5}$$