To simplify the given expression [tex]\(\frac{2 x^2 + 7 x + 3}{x^2 - 9}\)[/tex], we'll break it down into steps:
### Step 1: Factorize the numerator
The numerator is [tex]\(2 x^2 + 7 x + 3\)[/tex]. We factorize this quadratic expression.
1. Identify coefficients: [tex]\(a = 2\)[/tex], [tex]\(b = 7\)[/tex], and [tex]\(c = 3\)[/tex].
2. Factorize: Look for two numbers that multiply to [tex]\(a \cdot c = 2 \cdot 3 = 6\)[/tex] and add to [tex]\(b = 7\)[/tex].
- These numbers are 6 and 1 (6+1 = 7 and 6*1 = 6).
3. Rewrite the middle term:
[tex]\[
2 x^2 + 6 x + x + 3
\][/tex]
4. Factor by grouping:
[tex]\[
2 x(x + 3) + 1(x + 3)
\][/tex]
5. Factor out common factors:
[tex]\[
(2 x + 1)(x + 3)
\][/tex]
### Step 2: Factorize the denominator
The denominator is [tex]\(x^2 - 9\)[/tex], which is a difference of squares.
[tex]\[
x^2 - 9 = (x + 3)(x - 3)
\][/tex]
### Step 3: Simplify the expression
Now we have the expression in factored form:
[tex]\[
\frac{2 x^2 + 7 x + 3}{x^2 - 9} = \frac{(2 x + 1)(x + 3)}{(x + 3)(x - 3)}
\][/tex]
We can cancel out the common factor [tex]\((x + 3)\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{(2 x + 1)(x + 3)}{(x + 3)(x - 3)} = \frac{2 x + 1}{x - 3} \quad \text{for } x \neq -3, 3
\][/tex]
### Final Simplified Form
Thus, the simplified form of the given expression is [tex]\(\frac{2 x + 1}{x - 3}\)[/tex].
### Correct Answer
From the given choices, the correct answer is:
[tex]\[
\boxed{\frac{2 x + 1}{x - 3}}
\][/tex]
