1. Stating the problem: We need to verify if $$((x+y)(z^2+w^2))^2 = (x+y)^2 + 2(x+y)(z^2+w^2) + (z^2+w^2)^2$$ is true. 2. Expanding the left side: $$((x+y)(z^2+w^2))^2 = ((x+y)(z^2+w^2)) \times ((x+y)(z^2+w^2)) = (x+y)^2 (z^2+w^2)^2$$ 3. Expanding the right side as given: $$ (x+y)^2 + 2(x+y)(z^2+w^2) + (z^2+w^2)^2 $$ 4. Checking equality: The left side is a product squared: $$(x+y)^2 (z^2+w^2)^2$$ The right side is a sum: $$(x+y)^2 + 2(x+y)(z^2+w^2) + (z^2+w^2)^2$$ This matches the binomial expansion formula for $(a+b)^2 = a^2 + 2ab + b^2$ only if the left side were $(x+y + z^2 + w^2)^2$. 5. Conclusion: The given equality is false because the left side is $((x+y)(z^2+w^2))^2 = (x+y)^2 (z^2+w^2)^2$, which is a product squared, whereas the right side is the expansion of the square of a sum, $(x+y + z^2 + w^2)^2$. Hence, $$((x+y)(z^2+w^2))^2 \neq (x+y)^2 + 2(x+y)(z^2+w^2) + (z^2+w^2)^2.$$