Question

What is `((3x^2-x-2)/(2x^2+5x+2)) ((2x^2+3x+1)/(3x^2+5x+2))`?

Answer 1

The product is `(x - 1)/(x + 2), x!= -2, -1, -2/3, -1/2`

Explanation:

For problems like these, multiply before factoring.

`=> (3x^2 - 3x+ 2x - 2)/(2x^2 + 4x + x + 2) xx (2x^2 + 2x + x + 1)/(3x^2 + 3x + 2x + 2)`

`=> (3x(x - 1) + 2(x - 1))/(2x(x + 2) + 1(x + 2)) xx (2x(x + 1) + 1(x + 1))/(3x(x + 1) + 2(x + 1))`

`=>((3x + 2)(x - 1))/((2x + 1)(x + 2)) xx ((2x + 1)(x + 1))/((3x + 2)(x + 1))`

Cancelling using the property `a/a = 1`, we get:

`=>(x - 1)/(x + 2)`

We should also note the restrictions on the original expression. These are found by setting both denominators to `0` and solving for `x`.

`2x^2 + 5x + 2 = 0" AND "3x^2 + 5x + 2 = 0`

`(2x + 1)(x + 2) = 0" AND "(3x + 2)(x + 1) = 0"`

`x = -1/2 and -2" AND "x = -2/3 and -1`

Hence, our restrictions are `x != -2, -1, -2/3, -1/2`.

Hopefully this helps!