1. State the problem: Simplify the expression $$ (2a + 2b)^2 + (2a - 2b)^2 $$. 2. Expand each square term using the formula $$(x + y)^2 = x^2 + 2xy + y^2$$ and $$(x - y)^2 = x^2 - 2xy + y^2$$: $$(2a + 2b)^2 = (2a)^2 + 2 \cdot 2a \cdot 2b + (2b)^2 = 4a^2 + 8ab + 4b^2$$ $$(2a - 2b)^2 = (2a)^2 - 2 \cdot 2a \cdot 2b + (2b)^2 = 4a^2 - 8ab + 4b^2$$ 3. Add the two expanded expressions: $$4a^2 + 8ab + 4b^2 + 4a^2 - 8ab + 4b^2$$ 4. Combine like terms: $$4a^2 + 4a^2 = 8a^2$$ $$8ab - 8ab = 0$$ $$4b^2 + 4b^2 = 8b^2$$ 5. Final simplified result: $$8a^2 + 8b^2$$ So, $$ (2a + 2b)^2 + (2a - 2b)^2 = 8a^2 + 8b^2 $$.