1. Let's state the problem: Verify if the equation $$3ab^2+(2a-b)^2=4a(a-2b)+b^2(3a+1)$$ is correct. 2. Expand both sides to check equality. 3. Left side (LHS): $$3ab^2 + (2a - b)^2 = 3ab^2 + (2a)^2 - 2 \cdot 2a \cdot b + b^2 = 3ab^2 + 4a^2 - 4ab + b^2$$ 4. Right side (RHS): $$4a(a - 2b) + b^2(3a + 1) = 4a^2 - 8ab + 3ab^2 + b^2$$ 5. Simplify both sides: - LHS: $$4a^2 - 4ab + 3ab^2 + b^2$$ - RHS: $$4a^2 - 8ab + 3ab^2 + b^2$$ 6. Compare terms: - Both have $$4a^2$$, $$3ab^2$$, and $$b^2$$. - LHS has $$-4ab$$, RHS has $$-8ab$$. 7. Since $$-4ab \neq -8ab$$, the equation is not correct. Final answer: The equation is incorrect because the terms involving $$ab$$ do not match.