To solve the problem of finding [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] where
[tex]\[ f(x) = 2x^2 - 5x - 3 \][/tex]
[tex]\[ g(x) = 2x^2 + 5x + 2 \][/tex]
follow these steps:
1. Identify the functions: We are given two polynomial functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
[tex]\[
f(x) = 2x^2 - 5x - 3
\][/tex]
[tex]\[
g(x) = 2x^2 + 5x + 2
\][/tex]
2. Form the ratio [tex]\( \frac{f}{g} \)[/tex]: The task is to find the quotient of these two functions.
[tex]\[
\left(\frac{f}{g}\right)(x) = \frac{2x^2 - 5x - 3}{2x^2 + 5x + 2}
\][/tex]
3. Analyze the expression: We'll examine and simplify the expression if possible. In this case, the given form is already in its simplest. There is no apparent factorization or simplification that can be done to either the numerator [tex]\( 2x^2 - 5x - 3 \)[/tex] or the denominator [tex]\( 2x^2 + 5x + 2 \)[/tex] directly by common factors.
4. Conclusion: Therefore, the answer to the given problem is simply the fraction of the two given polynomials.
[tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is
[tex]\[
\frac{2x^2 - 5x - 3}{2x^2 + 5x + 2}
\][/tex]
5. Selection of the answer: Among the given options, the correct one matches this simplified form:
[tex]\[
\frac{2x^2 - 5x - 3}{2x^2 + 5x + 2}
\][/tex]
So, the resulting form corresponds to the option:
[tex]\[
\frac{2x^2 - 4x - 7}{2x^2 + 5x + 2}
\][/tex]
which is directly
[tex]\[
\frac{2x^2 - 5x - 3}{2x^2 + 5x + 2}
\][/tex]
Thus, the correct answer is the choice that exactly represents this simplified ratio of polynomials.
