To determine for which quotient [tex]\( x = 7 \)[/tex] is an excluded value, we need to ensure that no denominator in the expressions becomes zero when [tex]\( x = 7 \)[/tex]. Let's evaluate each option step-by-step.
### Option A
[tex]\[ \frac{x+7}{x^2+6x-7} \div \frac{7}{2x+14} \][/tex]
1. First fraction's denominator:
[tex]\[ x^2 + 6x - 7 \][/tex]
Substitute [tex]\( x = 7 \)[/tex]:
[tex]\[ (7)^2 + 6(7) - 7 = 49 + 42 - 7 = 84 \][/tex]
This is not zero.
2. Second fraction's denominator:
[tex]\[ 2x + 14 \][/tex]
Substitute [tex]\( x = 7 \)[/tex]:
[tex]\[ 2(7) + 14 = 14 + 14 = 28 \][/tex]
This is not zero.
Thus, [tex]\( x = 7 \)[/tex] does not make any denominator in option A equal to zero.
### Option B
[tex]\[ \frac{x^2-49}{3x+21} \div \frac{x^2+7x}{3x} \][/tex]
1. First fraction's denominator:
[tex]\[ 3x + 21 \][/tex]
Substitute [tex]\( x = 7 \)[/tex]:
[tex]\[ 3(7) + 21 = 21 + 21 = 42 \][/tex]
This is not zero.
2. Second fraction's denominator:
[tex]\[ 3x \][/tex]
Substitute [tex]\( x = 7 \)[/tex]:
[tex]\[ 3(7) = 21 \][/tex]
This is not zero.
Thus, [tex]\( x = 7 \)[/tex] does not make any denominator in option B equal to zero.
### Option C
[tex]\[ \frac{x-7}{x^2+4x-21} \div \frac{x^2+49}{x+7} \][/tex]
1. First fraction's denominator:
[tex]\[ x^2 + 4x - 21 \][/tex]
Substitute [tex]\( x = 7 \)[/tex]:
[tex]\[ (7)^2 + 4(7) - 21 = 49 + 28 - 21 = 56 \][/tex]
This is not zero.
2. Second fraction's denominator:
[tex]\[ x+7 \][/tex]
Substitute [tex]\( x = 7 \)[/tex]:
[tex]\[ 7 + 7 = 14 \][/tex]
This is not zero.
Thus, [tex]\( x = 7 \)[/tex] does not make any denominator in option C equal to zero.
### Option D
[tex]\[ \frac{7x}{x^2-10x+21} \div \frac{x+7}{7} \][/tex]
1. First fraction's denominator:
[tex]\[ x^2 - 10x + 21 \][/tex]
Substitute [tex]\( x = 7 \)[/tex]:
[tex]\[ (7)^2 - 10(7) + 21 = 49 - 70 + 21 = 0 \][/tex]
This is zero.
2. Second fraction's denominator:
[tex]\[ 7 \][/tex]
This is a constant and never zero.
Since [tex]\( x = 7 \)[/tex] makes the denominator [tex]\( x^2 - 10x + 21 \)[/tex] in the first fraction equal to zero, it is an excluded value for option D.
Therefore, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
