In the expansion of (x + 8)(6 - 3x), what is the coefficient of x in the resulting expression?

Introduction / Context:
This algebra question checks the ability to expand a simple product of two binomials and then identify the coefficient of a particular power of x. Understanding how each term in the expansion contributes to the final coefficient of x is important for polynomial manipulation and simplification problems.

Given Data / Assumptions:

• The expression is (x + 8)(6 - 3x).
• We need the coefficient of the first power of x in the expanded form.
• All variables are real numbers and basic algebraic rules apply.

Concept / Approach:
The standard method is to use the distributive property, often called FOIL (First, Outer, Inner, Last) for binomials. After expansion we collect like terms and then pick out the coefficient of x. It is important to track signs carefully because negative coefficients frequently appear when we multiply by terms like -3x.

Step-by-Step Solution:
Step 1: Expand the product (x + 8)(6 - 3x). Step 2: Multiply x by each term in the second bracket: x * 6 = 6x and x * (-3x) = -3x^2. Step 3: Multiply 8 by each term in the second bracket: 8 * 6 = 48 and 8 * (-3x) = -24x. Step 4: Combine all terms: -3x^2 + 6x - 24x + 48. Step 5: Simplify the x terms: 6x - 24x = -18x, so the expression becomes -3x^2 - 18x + 48. Step 6: The coefficient of x is -18.

Verification / Alternative check:
As a quick check, we can substitute x = 1 and x = 0 to confirm the linear term. For x = 1, evaluate (1 + 8)(6 - 3) = 9 * 3 = 27. From -3x^2 - 18x + 48 with x = 1 we get -3 - 18 + 48 = 27, which matches, supporting that the expansion is correct and thus the coefficient is correct.

Why Other Options Are Wrong:

• Option A: 18 is the magnitude of the coefficient but without the correct negative sign.
• Option B: 30 does not appear in the simplified expression and comes from incorrect multiplication.
• Option D: -30 would arise from mixing up terms or miscomputing 6x and -24x.
• Option E: 0 suggests there is no x term, which is not correct because the expansion clearly contains an x term.

Common Pitfalls:
Common mistakes include forgetting to multiply every term in the first bracket with every term in the second one or losing track of negative signs. Some learners also confuse the coefficient of x with the constant term or the coefficient of x^2. Writing each intermediate product clearly helps prevent these errors.

Final Answer:
Hence, the coefficient of x is -18.