To determine the correct verbal description for the expression [tex]\( x (2y + 4z)^2 \)[/tex], we need to analyze the components of the expression carefully.
Let's break down the expression step-by-step:
1. Identify the factors in the expression:
- The expression is [tex]\( x \)[/tex] multiplied by [tex]\( (2y + 4z)^2 \)[/tex].
2. Analyze the inner term:
- Inside the parentheses, we have [tex]\( 2y + 4z \)[/tex].
- Both [tex]\( y \)[/tex] and [tex]\( z \)[/tex] are variables that do not depend on [tex]\( x \)[/tex].
3. Apply the squaring operation:
- The term [tex]\( (2y + 4z)^2 \)[/tex] squares the entire expression inside the parentheses.
- Squaring [tex]\( 2y + 4z \)[/tex] does not introduce any dependence on [tex]\( x \)[/tex].
4. Combine the terms:
- The entire expression now is [tex]\( x \)[/tex] multiplied by the squared term [tex]\( (2y + 4z)^2 \)[/tex].
Given the above breakdown:
- The term [tex]\( 2y + 4z \)[/tex] inside the parentheses, and subsequently [tex]\( (2y + 4z)^2 \)[/tex], does not depend on [tex]\( x \)[/tex].
- Therefore, [tex]\( x \)[/tex] is being multiplied by a term that does not depend on [tex]\( x \)[/tex].
The most accurate verbal description would be:
Option A: the product of [tex]\( x \)[/tex] and a factor not depending on [tex]\( x \)[/tex].
Thus, the correct answer is 1 which corresponds to Option A.
