When the expression (x + 9)(6 - 4x) is expanded and simplified, what is the coefficient of x^2 in the resulting polynomial?

Introduction / Context:
Finding the coefficient of a specific power of x after expanding a product of binomials is a common algebra task. It appears often in school mathematics, aptitude tests and polynomial related questions. In this problem, you are asked to expand (x + 9)(6 - 4x), simplify the result, and extract the coefficient of x^2. This requires careful distribution of each term and basic collection of like terms.

Given Data / Assumptions:

• The expression is (x + 9)(6 - 4x).
• x is an algebraic variable representing a real number.
• We need only the coefficient of x^2 in the simplified form.
• Standard distributive law applies: (a + b)(c + d) = ac + ad + bc + bd.

Concept / Approach:
To find the coefficient of x^2, we expand the product term by term using the distributive property. After expansion, we combine like terms and identify the term that contains x^2. The coefficient is the numeric factor multiplying x^2. Observing which combinations of terms produce x^2 can sometimes speed up the process, because only products where the total power of x is two need to be considered.

Step-by-Step Solution:
Step 1: Write the product: (x + 9)(6 - 4x). Step 2: Distribute x across the second bracket: x * 6 gives 6x, and x * (-4x) gives -4x^2. Step 3: Distribute 9 across the second bracket: 9 * 6 gives 54, and 9 * (-4x) gives -36x. Step 4: Combine all terms: -4x^2 + 6x - 36x + 54. Step 5: Simplify like terms in x: 6x - 36x = -30x, so the expression becomes -4x^2 - 30x + 54. Step 6: The coefficient of x^2 in the simplified expression is therefore -4.

Verification / Alternative check:
You can verify the presence of the x^2 term by focusing only on the products that create x^2. These arise when x in the first bracket multiplies -4x in the second bracket, producing -4x^2. The constant 9 and the constant 6 do not contribute any x^2 term, nor do the products involving only one x. Since there is only one source of x^2, the coefficient must be -4. This quick reasoning supports the detailed expansion result.

Why Other Options Are Wrong:

• Option a (54) is the constant term, not the coefficient of x^2.
• Option c (30) is related to the coefficient of x after combining terms, but with the wrong sign and wrong power.
• Option d (4) has the correct magnitude but incorrect sign, which would arise if the negative sign on -4x were ignored.
• Option e (-30) is the coefficient of x, not of x^2, after simplification.

Common Pitfalls:
Students sometimes misapply the distributive law and miss one of the four products, or they mishandle signs, especially when dealing with negative coefficients. Another common issue is confusing coefficients of different powers of x. To avoid these mistakes, write down each term generated by the expansion, then carefully group like terms and check signs. Double checking the term that specifically contains x^2 ensures the correct coefficient is identified.

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