To determine which polynomial is equivalent to the expression
[tex]\[
\frac{2 x^2 + 3 x - 9}{(x + 3)(2 x + 3)},
\][/tex]
we need to simplify the given expression step by step.
First, we write down the numerator and the denominator:
- Numerator: [tex]\(2 x^2 + 3 x - 9\)[/tex]
- Denominator: [tex]\((x + 3) (2 x + 3)\)[/tex]
The numerator is a quadratic polynomial, and we need to see if it can be simplified or factored to match any of the options given in the question.
Comparing the numerator and denominator, we notice that we should check if the whole fraction can be simplified. To do this, we factorize if possible and simplify the expression.
It turns out that the given expression simplifies directly to:
[tex]\[
\frac{2 x^2 + 3 x - 9}{(x + 3)(2 x + 3)} = \frac{2 x - 3}{2 x + 3}
\][/tex]
Thus, after the simplification, the equivalent polynomial is:
[tex]\[
\frac{2 x - 3}{2 x + 3}
\][/tex]
So, the correct answer is:
A. [tex]\(\frac{2 z-3}{2 z+3}\)[/tex]
(Note: Here, [tex]\(z\)[/tex] is assumed to be a typographical error and it should be [tex]\(x\)[/tex] to match the variable in the original expression. Hence, consider the polynomial [tex]\( \frac{2 x - 3}{2 x + 3} \)[/tex].)
