To determine for which quotient the value [tex]\( x = 7 \)[/tex] is excluded, we need to check if [tex]\( x = 7 \)[/tex] makes any denominator in the given expressions equal to zero.
### Quotient A
[tex]\[
\frac{x-7}{x^2 + 4x - 21} \div \frac{x^2 + 49}{x + 7}
\][/tex]
For [tex]\( x = 7 \)[/tex]:
- The denominator of the first fraction: [tex]\( x^2 + 4x - 21 \)[/tex]
[tex]\[
7^2 + 4 \cdot 7 - 21 = 49 + 28 - 21 = 56 \quad (\text{not zero})
\][/tex]
- The denominator of the second fraction: [tex]\( x + 7 \)[/tex]
[tex]\[
7 + 7 = 14 \quad (\text{not zero})
\][/tex]
Conclusively, [tex]\( x = 7 \)[/tex] does not make any denominator zero in Quotient A.
### Quotient B
[tex]\[
\frac{x^2 - 49}{3x + 21} \div \frac{x^2 + 7x}{3x}
\][/tex]
For [tex]\( x = 7 \)[/tex]:
- The denominator of the first fraction: [tex]\( 3x + 21 \)[/tex]
[tex]\[
3 \cdot 7 + 21 = 21 + 21 = 42 \quad (\text{not zero})
\][/tex]
- The denominator of the second fraction: [tex]\( 3x \)[/tex]
[tex]\[
3 \cdot 7 = 21 \quad (\text{not zero})
\][/tex]
Conclusively, [tex]\( x = 7 \)[/tex] does not make any denominator zero in Quotient B.
### Quotient C
[tex]\[
\frac{x + 7}{x^2 + 6x - 7} \div \frac{7}{2x + 14}
\][/tex]
For [tex]\( x = 7 \)[/tex]:
- The denominator of the first fraction: [tex]\( x^2 + 6x - 7 \)[/tex]
[tex]\[
7^2 + 6 \cdot 7 - 7 = 49 + 42 - 7 = 84 \quad (\text{not zero})
\][/tex]
- The denominator of the second fraction: [tex]\( 2x + 14 \)[/tex]
[tex]\[
2 \cdot 7 + 14 = 14 + 14 = 28 \quad (\text{not zero})
\][/tex]
Conclusively, [tex]\( x = 7 \)[/tex] does not make any denominator zero in Quotient C.
### Quotient D
[tex]\[
\frac{7x}{x^2 - 10x + 21} \div \frac{x + 7}{7}
\][/tex]
For [tex]\( x = 7 \)[/tex]:
- The denominator of the first fraction: [tex]\( x^2 - 10x + 21 \)[/tex]
[tex]\[
7^2 - 10 \cdot 7 + 21 = 49 - 70 + 21 = 0 \quad (\text{zero})
\][/tex]
- The denominator of the second fraction: [tex]\( 7 \)[/tex] is a constant and does not depend on [tex]\( x \)[/tex].
Conclusively, [tex]\( x = 7 \)[/tex] makes the denominator of the first fraction zero in Quotient D.
Thus, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
