Respuesta :

[tex] \frac{2 {x}^{2} - 4x - 6}{x + 2} . \frac{ {x}^{2} - 4}{2 {x}^{2} + 2x } \\ \\ = \frac{2 {x}^{2} + 2x - 6x - 6}{x + 2} . \frac{ {x}^{2} - 4}{2x(x + 1) } \\ \\ = \frac{2x(x + 1) - 6(x + 1)}{x + 2} . \frac{ {x}^{2} - 4}{2x(x +1)} \\ \\ \frac{(2x - 6)(x + 1)}{x + 2} . \frac{ {x}^{2} - 4 }{2x(x + 1)} \\ \\ = \frac{2x - 6}{x + 2} . \frac{(x - 2)(x + 2)}{2x} \\ \\ = (2x - 6). \frac{x - 2}{2x} \\ \\ = \frac{2(x - 3)}{2x} .(x - 2) \\ \\ = \frac{x - 3}{x} .(x - 2)[/tex]

Option A)

carlosego

For this case we must simplify the following expression:

[tex]\frac{2x^2-4x-6}{x+2}*\frac{x^2-4}{2x^2+2x}[/tex]

So, by rewriting we have:

[tex]2x^2-4x-6=2(x^2-2x-3)[/tex]

We factor the parenthesis trinomial by looking for two numbers that, when multiplied, are obtained -3 and when added together, -2 is obtained. These numbers are -3 and 1, so:

[tex]2(x^2-2x-3)=2(x-3)(x+1)[/tex]

On the other hand we have to:

[tex]x ^ 2-4 = (x-2) (x + 2)[/tex]

Last we have:

[tex]2x ^ 2 + 2x = 2x (x + 1)[/tex]

Thus, rewriting the expression:

[tex]\frac {2 (x-3) (x + 1)} {x + 2} * \frac {(x-2) (x + 2)} {2x (x + 1)} =[/tex]

Simplifying:

[tex]\frac {(x-3) (x-2)} {x}[/tex]

Answer:

Option A

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