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:

  1. 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].

  2. Multiply the simplified expressions:

    Multiply the simplified expressions:
    [tex](2x + 1)(x - 1)[/tex]

  3. 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].