To solve for [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex], we need to divide the function [tex]\(f(x)\)[/tex] by the function [tex]\(g(x)\)[/tex]. Let's proceed step-by-step.
### Step 1: Define the functions
Given:
[tex]\[ f(x) = 4x^2 + 6x \][/tex]
[tex]\[ g(x) = 2x^2 + 13x + 15 \][/tex]
### Step 2: Form the fraction
We want to find the expression [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex]:
[tex]\[
\left( \frac{f}{g} \right)(x) = \frac{4x^2 + 6x}{2x^2 + 13x + 15}
\][/tex]
### Step 3: Simplify the fraction
To simplify this fraction, let us factor the terms where possible.
#### Numerator:
[tex]\[ f(x) = 4x^2 + 6x = 2x(2x + 3) \][/tex]
#### Denominator:
For the denominator, we'll factor [tex]\(2x^2 + 13x + 15\)[/tex]. This quadratic expression can be factored as follows:
[tex]\[ g(x) = 2x^2 + 13x + 15 = (2x + 3)(x + 5) \][/tex]
### Step 4: Rewrite the fraction with the factored forms
Substituting the factored forms into the fraction, we get:
[tex]\[
\frac{4x^2 + 6x}{2x^2 + 13x + 15} = \frac{2x(2x + 3)}{(2x + 3)(x + 5)}
\][/tex]
### Step 5: Cancel common factors in the numerator and denominator
Observe that both the numerator and the denominator have a common factor of [tex]\(2x + 3\)[/tex]. We can cancel this term out:
[tex]\[
\frac{2x(2x + 3)}{(2x + 3)(x + 5)} = \frac{2x \cancel{(2x + 3)}}{\cancel{(2x + 3)} (x + 5)} = \frac{2x}{x + 5}
\][/tex]
Hence, the simplified form of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is:
[tex]\[
\left( \frac{f}{g} \right)(x) = \frac{2x}{x + 5}
\][/tex]
### Final Answer
[tex]\[
\left( \frac{f}{g} \right)(x) = \frac{2x}{x + 5}
\][/tex]
