Which of the following is the correct algebraic expansion of (4x − 5y)^2? Choose the fully expanded expression with correct middle term and signs.

Introduction / Context: This question tests the standard binomial square expansion. The expression (a - b)^2 expands to a^2 - 2ab + b^2. A very common mistake is to forget the factor 2 in the middle term or to mishandle the sign. Here, a = 4x and b = 5y, so both the coefficient multiplication and the sign in the middle term must be handled correctly.

Given Data / Assumptions:

• Expression to expand: (4x - 5y)^2 • Required: fully expanded polynomial form

Concept / Approach: Use: (a - b)^2 = a^2 - 2ab + b^2. Substitute a = 4x and b = 5y: a^2 = (4x)^2, b^2 = (5y)^2, and 2ab = 2*(4x)*(5y). The sign of the middle term is negative because the binomial is (a - b).

Step-by-Step Solution: 1) Identify a and b: a = 4x, b = 5y 2) Compute a^2: a^2 = (4x)^2 = 16x^2 3) Compute 2ab: 2ab = 2*(4x)*(5y) = 40xy 4) Compute b^2: b^2 = (5y)^2 = 25y^2 5) Combine with correct signs: (4x - 5y)^2 = 16x^2 - 40xy + 25y^2

Verification / Alternative check: Pick simple values x = 1, y = 1: LHS: (4 - 5)^2 = (-1)^2 = 1. RHS using correct expansion: 16 - 40 + 25 = 1. So the expansion is consistent and verified.

Why Other Options Are Wrong: • -20xy misses the required factor 2 in the middle term. • +40xy has the wrong sign (would correspond to (4x + 5y)^2). • -25y^2 has the wrong sign; squares are always non-negative in expansion. • +20xy is both wrong sign and wrong coefficient.

Common Pitfalls: • Forgetting the 2 in 2ab. • Treating (a - b)^2 as a^2 - b^2 (incorrect).

Final Answer: 16x^2 - 40xy + 25y^2