Let's analyze the expression given in the problem:
[tex]\[
x(2y + 4z)^2
\][/tex]
We need to select the correct verbal description for this expression from the provided options.
First, note that the expression involves two main parts:
1. [tex]\( x \)[/tex]
2. [tex]\( (2y + 4z)^2 \)[/tex]
The entire expression can be described as a product because the main operation is multiplication of [tex]\( x \)[/tex] and [tex]\( (2y + 4z)^2 \)[/tex].
Now, let's break down each component:
- [tex]\( x \)[/tex] is a variable by itself.
- [tex]\( (2y + 4z)^2 \)[/tex] is another factor that is to be squared. However, this factor does not depend on [tex]\( x \)[/tex].
To form the complete expression [tex]\( x(2y + 4z)^2 \)[/tex], we are multiplying [tex]\( x \)[/tex] by the squared term [tex]\( (2y + 4z)^2 \)[/tex]. Observe that the factor [tex]\( (2y + 4z)^2 \)[/tex] is independent of [tex]\( x \)[/tex]; it only depends on variables [tex]\( y \)[/tex] and [tex]\( z \)[/tex].
Based on this analysis, the expression matches the description where [tex]\( x \)[/tex] is multiplied by another factor that does not depend on [tex]\( x \)[/tex].
Therefore, the correct verbal description is:
[tex]\[
\boxed{C. \text{the product of } x \text{ and a factor not depending on } x}
\][/tex]
