Certainly! Let's break down the given verbal description step-by-step, and then identify the correct mathematical expression from the given options.
### Verbal Description:
"the square of the sum of 7 times [tex]\( x \)[/tex] and 3 divided by 4 times the difference of [tex]\( x \)[/tex] and 1"
### Step-by-Step Breakdown:
1. Sum of 7 times [tex]\( x \)[/tex] and 3:
- This can be written as:
[tex]\[
7x + 3
\][/tex]
2. Square of this sum:
- Squaring the expression [tex]\( 7x + 3 \)[/tex], we get:
[tex]\[
(7x + 3)^2
\][/tex]
3. Difference of [tex]\( x \)[/tex] and 1:
- This can be written as:
[tex]\[
x - 1
\][/tex]
4. 4 times this difference:
- Multiplying [tex]\( x - 1 \)[/tex] by 4, we get:
[tex]\[
4(x - 1)
\][/tex]
5. Entire Expression:
- Now we need to take the square of the sum [tex]\( (7x + 3)^2 \)[/tex] and divide it by [tex]\( 4(x - 1) \)[/tex]:
[tex]\[
\frac{(7x + 3)^2}{4(x - 1)}
\][/tex]
### Matching with Options:
Let's compare this derived expression with the given options:
A. [tex]\(\frac{7 x^2+3}{4 x-1}\)[/tex]
- This expression does not match our derived expression. It misses the square on [tex]\( 7x + 3 \)[/tex] and the correct form of the divisor.
B. [tex]\(\frac{(7 x+3)^2}{4 x-1}\)[/tex]
- This expression is close, but the denominator is incorrect. It should be [tex]\( 4(x - 1) \)[/tex], not [tex]\( 4x - 1 \)[/tex].
C. [tex]\(\frac{(7 x+3)^2}{4(x-1)}\)[/tex]
- This expression matches our derived expression exactly. The numerator is [tex]\( (7x + 3)^2 \)[/tex] and the denominator is [tex]\( 4(x - 1) \)[/tex].
D. [tex]\(\frac{7 x^2+3}{4(x-1)}\)[/tex]
- This expression does not match our derived expression. The numerator is incorrect.
Given these comparisons, the correct answer is:
### Final Answer:
C. [tex]\(\frac{(7 x+3)^2}{4(x-1)}\)[/tex]
Therefore, the expression [tex]\(\frac{(7 x+3)^2}{4(x-1)}\)[/tex] correctly represents the verbal description provided.
