1. **Problem statement:** Expand and reduce the polynomial $$Q(x) = (4x + 5)^2 - (x - 2)(4x + 5)$$. 2. **Recall formulas:** - Square of a binomial: $$(a + b)^2 = a^2 + 2ab + b^2$$ - Distributive property: $$a(b + c) = ab + ac$$ 3. **Expand each term:** - Expand $$(4x + 5)^2$$: $$ (4x)^2 + 2 \cdot 4x \cdot 5 + 5^2 = 16x^2 + 40x + 25 $$ - Expand $$(x - 2)(4x + 5)$$: $$ x \cdot 4x + x \cdot 5 - 2 \cdot 4x - 2 \cdot 5 = 4x^2 + 5x - 8x - 10 = 4x^2 - 3x - 10 $$ 4. **Substitute expansions back into $Q(x)$:** $$ Q(x) = (16x^2 + 40x + 25) - (4x^2 - 3x - 10) $$ 5. **Simplify by distributing the minus sign and combining like terms:** $$ Q(x) = 16x^2 + 40x + 25 - 4x^2 + 3x + 10 = (16x^2 - 4x^2) + (40x + 3x) + (25 + 10) $$ $$ Q(x) = 12x^2 + 43x + 35 $$ **Final answer:** $$\boxed{Q(x) = 12x^2 + 43x + 35}$$