To simplify the expression [tex]\(\left(\frac{2 x^2 + 5 x + 2}{x + 1}\right) \left(\frac{x^2 - 1}{x + 2}\right)\)[/tex], we need to perform the following steps:
1. Factorize the Numerators and Denominators:
- For [tex]\(\frac{2x^2 + 5x + 2}{x + 1}\)[/tex]:
We can factorize [tex]\(2x^2 + 5x + 2\)[/tex] as follows:
[tex]\[
2x^2 + 5x + 2 = (2x + 1)(x + 2)
\][/tex]
So,
[tex]\[
\frac{2x^2 + 5x + 2}{x + 1} = \frac{(2x + 1)(x + 2)}{x + 1}
\][/tex]
- For [tex]\(\frac{x^2 - 1}{x + 2}\)[/tex]:
We can factorize [tex]\(x^2 - 1\)[/tex] as follows:
[tex]\[
x^2 - 1 = (x - 1)(x + 1)
\][/tex]
So,
[tex]\[
\frac{x^2 - 1}{x + 2} = \frac{(x - 1)(x + 1)}{x + 2}
\][/tex]
2. Simplify the Product of the Expressions:
Using the factored forms, we can write:
[tex]\[
\left(\frac{(2x + 1)(x + 2)}{x + 1}\right)\left(\frac{(x - 1)(x + 1)}{x + 2}\right)
\][/tex]
Next, we cancel the common terms in the numerator and the denominator:
- The term [tex]\(x + 2\)[/tex] appears in both the numerator and denominator.
- The term [tex]\(x + 1\)[/tex] appears in both the numerator and denominator.
Therefore, the expression simplifies to:
[tex]\[
(2x + 1)(x - 1)
\][/tex]
3. Expand the Simplified Expression:
Now, expand [tex]\((2x + 1)(x - 1)\)[/tex]:
[tex]\[
(2x + 1)(x - 1) = 2x(x - 1) + 1(x - 1)
\][/tex]
[tex]\[
= 2x^2 - 2x + x - 1
\][/tex]
[tex]\[
= 2x^2 - x - 1
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
2x^2 - x - 1
\][/tex]
The correct answer is:
[tex]\[
\boxed{2x^2 - x - 1}
\][/tex]
