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. Identify the expressions in the numerators and denominators:
- Numerator of the first fraction: [tex]\(2x^2 + 5x + 2\)[/tex]
- Denominator of the first fraction: [tex]\(x + 1\)[/tex]
- Numerator of the second fraction: [tex]\(x^2 - 1\)[/tex]
- Denominator of the second fraction: [tex]\(x + 2\)[/tex]
2. Factorize the numerators and denominators if possible:
- [tex]\(2x^2 + 5x + 2\)[/tex] can be factored as [tex]\((2x+1)(x+2)\)[/tex]
- [tex]\(x^2 - 1\)[/tex] is a difference of squares, which can be factored as [tex]\((x+1)(x-1)\)[/tex]
3. Rewrite the expression using these factorizations:
[tex]\[
\left(\frac{(2x+1)(x+2)}{x+1}\right) \left(\frac{(x+1)(x-1)}{x+2}\right)
\][/tex]
4. Simplify by canceling common factors in the numerators and denominators:
- The [tex]\((x + 2)\)[/tex] in the numerator of the first fraction and the denominator of the second fraction cancel out.
- The [tex]\((x + 1)\)[/tex] in the numerator of the second fraction and the denominator of the first fraction cancel out.
After canceling the common factors, we are left with:
[tex]\[
(2x+1)(x-1)
\][/tex]
5. Multiply the simplified expressions:
[tex]\[
(2x+1)(x-1)
\][/tex]
Expanding the product gives:
[tex]\[
(2x+1)(x-1) = 2x(x) + 2x(-1) + 1(x) + 1(-1) = 2x^2 - 2x + x - 1 = 2x^2 - x - 1
\][/tex]
So, the simplified form of the given expression is:
[tex]\[
2x^2 - x - 1
\][/tex]
The correct answer is:
A) [tex]\(2x^2 - x - 1\)[/tex]
