`(x^2+6)/(x^2+2x-24)`

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You need to reduce the fraction to its lowest terms, hence, you need to decompose both numerator and denominator in linear factors.

You need to solve for x the following equations, such that:

`x^2 + 6 = 0 => x_(1,2) = sqrt(-6) => x_(1,2) = +-i*sqrt 6`

You need to write the factored form of numerator, such that:

`x^2 + 6 = (x - i*sqrt6)(x + i*sqrt6)`

`x^2 + 2x - 24 = 0`

Using quadratic formula yields:

`x_(1,2) = (-2+-sqrt(4 + 96))/2 =>x_(1,2) = (-2+-sqrt100)/2`

`x_(1,2) = (-2+-10)/2 => x_1 = 4 ; x_2 = -6`

You need to write the factored form of denominator, such that:

`x^2 + 2x - 24 = (x - 4)(x + 6)`

Since the given form of the fraction is `(x^2 + 6)/(x^2 + 2x - 24)` this fraction can be converted in its factored form, such that:

`(x^2 + 6)/(x^2 + 2x - 24) = ((x - i*sqrt6)(x + i*sqrt6))/((x - 4)(x + 6))`

You can notice that there exists no common factors, hence, the fraction cannot be reduced to its lowest terms.

If you intend to simplify the given fraction, then you need to correct the numerator that must be `x + 6` an not `x^2 + 6` .

If the numerator is the binomial `x + 6` , yields:

`(x + 6)/(x^2 + 2x - 24) = (x + 6)/((x - 4)(x + 6))`

Reducing duplicate factors, yields:

`(x + 6)/(x^2 + 2x - 24) = 1/(x - 4)`

Hence, reducing the fraction `(x + 6)/(x^2 + 2x - 24)` to its lowest terms, yields` (x + 6)/(x^2 + 2x - 24) = 1/(x - 4).`

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