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], follow these steps:
1. Factor the numerator expressions:
- The numerator of the first fraction is [tex]\(2 x^2 + 5 x + 2\)[/tex].
To factor it, we look for two numbers that multiply to [tex]\(2 \cdot 2 = 4\)[/tex] and add to [tex]\(5\)[/tex]. These numbers are [tex]\(4\)[/tex] and [tex]\(1\)[/tex].
[tex]\[
2 x^2 + 5 x + 2 = (2 x + 1)(x + 2)
\][/tex]
- The numerator of the second fraction is [tex]\(x^2 - 1\)[/tex], which is a difference of squares.
[tex]\[
x^2 - 1 = (x - 1)(x + 1)
\][/tex]
2. Rewrite the given expression using the factored forms:
[tex]\[
\frac{(2 x + 1)(x + 2)}{x + 1} \cdot \frac{(x - 1)(x + 1)}{x + 2}
\][/tex]
3. Simplify by canceling common factors:
- The [tex]\((x + 1)\)[/tex] terms cancel out:
[tex]\[
\frac{(2 x + 1)}{1} \cdot \frac{(x - 1)}{1} = (2 x + 1)(x - 1)
\][/tex]
4. Perform the multiplication of the remaining factors:
- Multiply the binomials:
[tex]\[
(2 x + 1)(x - 1) = 2 x (x) + 2 x (-1) + 1 (x) + 1 (-1)
\][/tex]
[tex]\[
= 2 x^2 - 2 x + x - 1
\][/tex]
[tex]\[
= 2 x^2 - x - 1
\][/tex]
Thus, the simplified form of the given expression is [tex]\(2 x^2 - x - 1\)[/tex].
Therefore, the correct answer is:
A. [tex]\(2 x^2 - x - 1\)[/tex].
