Question

How do you simplify `(6x^2-4x-3)/(3x^2+x)`?

Answer 1

`(6x^2-4x-3)/(3x^2+x) = 2-(6x+3)/(3x^2+x)`

Explanation:

Notice that:

`3x^2+x = x(3x+1)`

Then `x` is not a factor of the numerator, but `(3x+1)` might be.

Let's try:

`6x^2-4x-3 = (6x^2+2x)-(6x+2)-1`

`color(white)(6x^2-4x-3) = 2x(3x+1)-2(3x+1)-1`

`color(white)(6x^2-4x-3) = (2x-2)(3x+1)-1`

No. So there is no cancellable factor and we cannot substantially simplify the given rational expression.

About the most we can do is separate out the quotient `2` as follows:

`(6x^2-4x-3)/(3x^2+x) = ((6x^2+2x)-(6x+3))/(3x^2+x)`

`color(white)((6x^2-4x-3)/(3x^2+x)) = (2(3x^2+x)-(6x+3))/(3x^2+x)`

`color(white)((6x^2-4x-3)/(3x^2+x)) = 2-(6x+3)/(3x^2+x)`