Answer :
To determine the product of the given rational expressions, let's simplify the expression step-by-step.
The given expression is:
[tex]\[ \frac{x+10}{x+3} \cdot \frac{x-10}{x-3} \][/tex]
### Step 1: Multiply the numerators
Multiply the numerators together:
[tex]\[ (x+10) \cdot (x-10) \][/tex]
This results in the difference of squares:
[tex]\[ (x+10)(x-10) = x^2 - 10^2 = x^2 - 100 \][/tex]
### Step 2: Multiply the denominators
Multiply the denominators together:
[tex]\[ (x+3) \cdot (x-3) \][/tex]
This also results in the difference of squares:
[tex]\[ (x+3)(x-3) = x^2 - 3^2 = x^2 - 9 \][/tex]
### Step 3: Combine the simplified numerator and denominator
Now, we place the simplified numerator and denominator together as one rational expression:
[tex]\[ \frac{x^2 - 100}{x^2 - 9} \][/tex]
Hence, the product of the given rational expressions is:
[tex]\[ \boxed{\frac{x^2 - 100}{x^2 - 9}} \][/tex]
Therefore, the correct answer is:
D. [tex]\(\frac{x^2 - 100}{x^2 - 9}\)[/tex]
The given expression is:
[tex]\[ \frac{x+10}{x+3} \cdot \frac{x-10}{x-3} \][/tex]
### Step 1: Multiply the numerators
Multiply the numerators together:
[tex]\[ (x+10) \cdot (x-10) \][/tex]
This results in the difference of squares:
[tex]\[ (x+10)(x-10) = x^2 - 10^2 = x^2 - 100 \][/tex]
### Step 2: Multiply the denominators
Multiply the denominators together:
[tex]\[ (x+3) \cdot (x-3) \][/tex]
This also results in the difference of squares:
[tex]\[ (x+3)(x-3) = x^2 - 3^2 = x^2 - 9 \][/tex]
### Step 3: Combine the simplified numerator and denominator
Now, we place the simplified numerator and denominator together as one rational expression:
[tex]\[ \frac{x^2 - 100}{x^2 - 9} \][/tex]
Hence, the product of the given rational expressions is:
[tex]\[ \boxed{\frac{x^2 - 100}{x^2 - 9}} \][/tex]
Therefore, the correct answer is:
D. [tex]\(\frac{x^2 - 100}{x^2 - 9}\)[/tex]
