Simplify (x^2+12x+12)/(x^3-4x)

Problem

(x2+12*x+12)/(x3−4*x)

Solution

1. Factor the denominator by taking out the greatest common factor, which is x

(x2+12*x+12)/(x*(x2−4))

2. Recognize the difference of squares in the denominator, where x2−4 can be factored into (x−2)*(x+2)

(x2+12*x+12)/(x*(x−2)*(x+2))

3. Check if the numerator x2+12*x+12 can be factored to cancel any terms with the denominator. The factors of 12 that add up to 12 do not exist among integers (the discriminant 12−4*(1)*(12)=96 is not a perfect square).

4. Conclude that since no factors are common between the numerator and the denominator, the expression is in its simplest factored form.

Final Answer

(x2+12*x+12)/(x3−4*x)=(x2+12*x+12)/(x*(x−2)*(x+2))

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