Answer :
To simplify and find the product of the given expressions [tex]\frac{2x^2 + 5x + 2}{x + 1}[/tex] and [tex]\frac{x^2 - 1}{x + 2}[/tex], we can follow these steps:
Simplify each expression separately:
For the first expression, [tex]\frac{2x^2 + 5x + 2}{x + 1}[/tex], we perform polynomial division or factor the numerator directly:
- The numerator [tex]2x^2 + 5x + 2[/tex] can be factored into [tex](2x + 1)(x + 2)[/tex].
- Therefore, [tex]\frac{2x^2 + 5x + 2}{x + 1} = \frac{(2x + 1)(x + 2)}{x + 1} = 2x + 1[/tex] when [tex]x \neq -1[/tex].
For the second expression, [tex]\frac{x^2 - 1}{x + 2}[/tex], factor the numerator:
- The numerator [tex]x^2 - 1[/tex] can be factored into [tex](x - 1)(x + 1)[/tex].
- Therefore, [tex]\frac{x^2 - 1}{x + 2} = \frac{(x - 1)(x + 1)}{x + 2} = x - 1[/tex] when [tex]x \neq -2[/tex].
Multiply the simplified expressions:
Multiply the simplified expressions:
[tex](2x + 1)(x - 1)[/tex]Expand the product expression:
Expanding [tex](2x + 1)(x - 1)[/tex]:
[tex](2x + 1)(x - 1) = 2x^2 - 2x + x - 1 = 2x^2 - x - 1[/tex]
So, the product of the expressions [tex]\frac{2x^2 + 5x + 2}{x + 1}[/tex] and [tex]\frac{x^2 - 1}{x + 2}[/tex] simplifies to [tex]2x^2 - x - 1[/tex].
Thus, the correct option is:
[tex]\boxed{2x^2 - x - 1}[/tex]
The answer is [tex]A) 2x^2 - x - 1[/tex].
