Sure! Let's find the quotient of the rational expressions step by step and simplify it.
Given the rational expressions:
[tex]\[ \frac{x^2 - 9}{x + 1} \][/tex]
and
[tex]\[ \frac{x^2 - 6x + 9}{7x + 7} \][/tex]
### Step 1: Factoring the numerators
1. For the first expression, the numerator [tex]\( x^2 - 9 \)[/tex] can be factored as [tex]\( (x - 3)(x + 3) \)[/tex].
[tex]\[ x^2 - 9 = (x - 3)(x + 3) \][/tex]
So we have:
[tex]\[ \frac{(x - 3)(x + 3)}{x + 1} \][/tex]
2. For the second expression, the numerator [tex]\( x^2 - 6x + 9 \)[/tex] can be factored as [tex]\( (x - 3)^2 \)[/tex].
[tex]\[ x^2 - 6x + 9 = (x - 3)^2 \][/tex]
### Step 2: Factoring the denominators
1. The denominator of the first expression, [tex]\( x + 1 \)[/tex], cannot be factored further.
2. The denominator [tex]\( 7x + 7 \)[/tex] of the second expression can be factored as [tex]\( 7(x + 1) \)[/tex].
[tex]\[ 7x + 7 = 7(x + 1) \][/tex]
Thus, we have:
[tex]\[ \frac{(x - 3)(x + 3)}{x + 1} \][/tex]
and
[tex]\[ \frac{(x - 3)^2}{7(x + 1)} \][/tex]
### Step 3: Combining the expressions
The quotient of the two rational expressions is given by:
[tex]\[ \left( \frac{(x - 3)(x + 3)}{x + 1} \right) \div \left( \frac{(x - 3)^2}{7(x + 1)} \right) \][/tex]
Dividing one fraction by another is equivalent to multiplying by the reciprocal:
[tex]\[ \left( \frac{(x - 3)(x + 3)}{x + 1} \right) \times \left( \frac{7(x + 1)}{(x - 3)^2} \right) \][/tex]
### Step 4: Simplifying the expression
Multiply the numerators and denominators:
[tex]\[ \frac{(x - 3)(x + 3) \cdot 7(x + 1)}{(x + 1) \cdot (x - 3)^2} \][/tex]
Then cancel common factors in the numerator and the denominator:
1. [tex]\( x + 1 \)[/tex] cancels out,
2. One [tex]\( (x - 3) \)[/tex] in the numerator cancels with one [tex]\( (x - 3) \)[/tex] in the denominator
Thus, we have:
[tex]\[ \frac{7(x + 3)}{(x - 3)} \][/tex]
Hence, the quotient of the given rational expressions in reduced form is:
[tex]\[ \boxed{\frac{7(x + 3)}{x - 3}} \][/tex]
This matches option [tex]\( B. \)[/tex]
