Find the positive, negative and complex roots of the following function f(x)=x^6-x^5-12x-x^4+4x^3-12x^2+12

1) That equation cannot be factored. And you cannot find roots by using the root theorem or any exact method.

2)  That means that you should use either a numerical method, which exceeds the high school level (even the basics college levels), or a graphing calculator.

3) I opted for the graphing calculator and found these roots for you:

One important observation is that the degree of polynomila, i.e. 6, indicate the maximum number of roots:

So, you can expect 6 solutions. These are:

Real roots: 
x ≈ 0.65
x ≈ 2.13

Complex roots:

x ≈ -1.48 - 0.24i
x ≈ - 1.48 + 0.24i
x ≈ 0.59 - 1.87i
x ≈ 0.59 + 1.87i