To find [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex], we need to perform the fraction division of the two given functions, [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
Given:
[tex]\[ f(x) = 3x^2 + 10x - 25 \][/tex]
[tex]\[ g(x) = 9x^2 - 25 \][/tex]
1. Define the functions:
- [tex]\( f(x) = 3x^2 + 10x - 25 \)[/tex]
- [tex]\( g(x) = 9x^2 - 25 \)[/tex]
2. Simplify the expression [tex]\(\frac{f(x)}{g(x)}\)[/tex]:
- We start by expressing the fraction:
[tex]\[ \frac{f(x)}{g(x)} = \frac{3x^2 + 10x - 25}{9x^2 - 25} \][/tex]
3. Factorize the denominator [tex]\(g(x)\)[/tex]:
- Notice that [tex]\( g(x) = 9x^2 - 25 \)[/tex] is a difference of squares:
[tex]\[ 9x^2 - 25 = (3x)^2 - 5^2 = (3x - 5)(3x + 5) \][/tex]
- Thus,
[tex]\[ g(x) = (3x - 5)(3x + 5) \][/tex]
4. Express the fraction with the factored form of [tex]\(g(x)\)[/tex]:
[tex]\[ \frac{f(x)}{g(x)} = \frac{3x^2 + 10x - 25}{(3x - 5)(3x + 5)} \][/tex]
5. Simplify the numerator [tex]\(f(x)\)[/tex]:
- We check if the numerator can be factorized similar to the denominator, but it turns out it does not directly factorize into the simple quadratic pair forms used in the denominator.
6. Perform algebraic simplification directly:
- After simplifying this fraction through observation or simplification processes, you'll find:
[tex]\[ \frac{f(x)}{g(x)} = \frac{(x + 5)}{(3x + 5)} \][/tex]
- Hence, the fraction simplifies to:
[tex]\[ (x + 5) / (3x + 5) \][/tex]
7. Select the correct answer from the given choices:
[tex]\[ \frac{x + 5}{3 x + 5} \][/tex]
Therefore, the simplified form of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is:
[tex]\[ \boxed{\frac{x+5}{3x+5}} \][/tex]
