To determine for which quotient [tex]\( x = 7 \)[/tex] is an excluded value, we need to evaluate the quotient's denominators when [tex]\( x = 7 \)[/tex] and check if any of them equal zero since division by zero is undefined.
### Option A: [tex]\(\frac{x^2-49}{3x+21} \div \frac{x^2+7x}{3x}\)[/tex]
1. Denominator 1: [tex]\( 3x + 21 \)[/tex]
[tex]\[
3(7) + 21 = 21 + 21 = 42 \quad (\text{Not zero})
\][/tex]
2. Denominator 2: [tex]\( 3x \)[/tex]
[tex]\[
3(7) = 21 \quad (\text{Not zero})
\][/tex]
### Option B: [tex]\(\frac{x-7}{x^2+4x-21} \div \frac{x^2+49}{x+7}\)[/tex]
1. Denominator 1: [tex]\( x^2 + 4x - 21 \)[/tex]
[tex]\[
(7)^2 + 4(7) - 21 = 49 + 28 - 21 = 56 \quad (\text{Not zero})
\][/tex]
2. Denominator 2: [tex]\( x + 7 \)[/tex]
[tex]\[
7 + 7 = 14 \quad (\text{Not zero})
\][/tex]
### Option C: [tex]\(\frac{x+7}{x^2+6x-7} \div \frac{7}{2x+14}\)[/tex]
1. Denominator 1: [tex]\( x^2 + 6x - 7 \)[/tex]
[tex]\[
(7)^2 + 6(7) - 7 = 49 + 42 - 7 = 84 \quad (\text{Not zero})
\][/tex]
2. Denominator 2: [tex]\( 2x + 14 \)[/tex]
[tex]\[
2(7) + 14 = 14 + 14 = 28 \quad (\text{Not zero})
\][/tex]
### Option D: [tex]\(\frac{7x}{x^2-10x+21} \div \frac{x+7}{7}\)[/tex]
1. Denominator 1: [tex]\( x^2 - 10x + 21 \)[/tex]
[tex]\[
(7)^2 - 10(7) + 21 = 49 - 70 + 21 = 0 \quad (\text{Zero})
\][/tex]
2. Denominator 2: [tex]\( 7 \)[/tex]
[tex]\[
(\text{This is a constant and does not affect the exclusion.})
\][/tex]
Since the denominator [tex]\( x^2 - 10x + 21 \)[/tex] evaluates to zero when [tex]\( x = 7 \)[/tex] in Option D, [tex]\( x = 7 \)[/tex] is an excluded value for Option D.
Therefore, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
