Find the coefficient of x^2 after expanding and simplifying the polynomial: (x + 9)(6 - 4x)(4x - 7).

Introduction / Context:
This algebra question asks you to find the coefficient of x^2 in the expansion of a product of three binomials. Instead of fully expanding to get every term, you can strategically multiply and track only the contributions to the x^2 term. This technique is useful in polynomial algebra, where full expansion can be time consuming, especially in aptitude tests with limited time.

Given Data / Assumptions:

• Expression: (x + 9)(6 − 4x)(4x − 7).
• We must expand and simplify, then identify the coefficient of x^2.
• x is a real variable.

Concept / Approach:
First, multiply the two binomials (6 − 4x) and (4x − 7) to get a quadratic in x. Then multiply that quadratic by (x + 9). During the second multiplication, we can focus on terms that produce x^2 in the final product. This reduces the amount of work compared with finding the entire expanded expression. Collecting all x^2 contributions yields the desired coefficient directly.

Step-by-Step Solution:
1) Multiply (6 − 4x)(4x − 7) first. 2) Compute 6 * (4x − 7) = 24x − 42. 3) Compute −4x * (4x − 7) = −16x^2 + 28x. 4) Add these to get (6 − 4x)(4x − 7) = −16x^2 + (24x + 28x) − 42 = −16x^2 + 52x − 42. 5) Now multiply (x + 9) by this quadratic: (x + 9)(−16x^2 + 52x − 42). 6) Multiply x by the quadratic: x * (−16x^2 + 52x − 42) = −16x^3 + 52x^2 − 42x. 7) Multiply 9 by the quadratic: 9 * (−16x^2 + 52x − 42) = −144x^2 + 468x − 378. 8) Combine like terms. For x^2, the total coefficient is 52x^2 − 144x^2 = −92x^2. 9) Therefore, the coefficient of x^2 in the full expansion is −92.

Verification / Alternative check:
As a check, you can use a symbolic expansion approach or re multiply while tracking terms more briefly. Since any x^2 term in the final product arises either from x times the x^2 term in the quadratic or from 9 times the x^2 term in that quadratic, our calculation already accounts for all such contributions. We do not get x^2 terms from combinations that involve three x factors (which produce x^3) or constant terms. Carefully re checking the intermediate quadratic and both partial products confirms the coefficient −92 is consistent.

Why Other Options Are Wrong:
Options a (216) and b (108) are large positive numbers and can arise from mis tracking x^3 or constant terms, but they do not match the correct combined x^2 coefficient. Option d (−4) and option e (92) result from partial or sign errors, such as using 52 − 48 or incorrectly adding absolute values. Only −92 is consistent with correct multiplication and term collection.

Common Pitfalls:
A common error is to expand all three factors at once or to multiply in a rushed manner, leading to sign mistakes or missing terms. Some students also forget that x times 52x contributes to x^2, while 9 times −16x^2 also contributes, and they may overlook one of these. Breaking the problem into two clear multiplication steps and systematically collecting only x^2 terms helps prevent these mistakes.

Final Answer:
The coefficient of x^2 in the expansion of (x + 9)(6 − 4x)(4x − 7) is -92.