Find the expanded square of the binomial expression (7 − 4x) and write it as a standard quadratic expression in x.

Introduction / Context: This question is about expanding the square of a binomial, a very common algebra skill. Recognising and applying the identity for (a − b)^2 or (a + b)^2 allows you to quickly convert a compact expression into a standard polynomial form, which is heavily used in solving quadratic equations and simplifying algebraic expressions.

Given Data / Assumptions:

• The binomial is (7 − 4x).
• We need to compute (7 − 4x)^2.
• The answer should be written as a quadratic expression in x.

Concept / Approach: Main identity:

• (a − b)^2 = a^2 − 2ab + b^2.
• Here, a = 7 and b = 4x.
• Compute each term carefully: a^2, 2ab, and b^2.

Step-by-Step Solution: Write the binomial square: (7 − 4x)^2. Identify a = 7 and b = 4x. Compute a^2 = 7^2 = 49. Compute 2ab = 2 * 7 * 4x = 56x. Compute b^2 = (4x)^2 = 16x^2. Apply the identity: (7 − 4x)^2 = a^2 − 2ab + b^2. So (7 − 4x)^2 = 49 − 56x + 16x^2. Rearrange into standard quadratic order: 16x^2 − 56x + 49.

Verification / Alternative check: You can expand directly by multiplication: (7 − 4x)(7 − 4x). Multiply term by term: 7 * 7 = 49, 7 * (−4x) = −28x, (−4x) * 7 = −28x, and (−4x) * (−4x) = 16x^2. Adding these gives 49 − 56x + 16x^2, which matches the result from the identity.

Why Other Options Are Wrong: Option a: 16x^2 − 28x + 49 uses 28x instead of 56x, forgetting that the middle term is twice the product ab. Option b and option c: These forms either have incorrect coefficients or wrong signs for the quadratic term and do not match the proper expansion. Option e: 49 + 56x + 16x^2 corresponds to (7 + 4x)^2, not (7 − 4x)^2, so the middle term has the wrong sign.

Common Pitfalls: A frequent error is to forget the factor of 2 in the middle term or to mis-handle the sign when the binomial includes a minus. Carefully applying the identity and double checking the sign of 2ab prevents such mistakes.

Final Answer: The expanded quadratic expression is 16x^2 − 56x + 49.