Step 1 :Compute the derivatives of the functions in the numerator and denominator: \(f'(x) = 3x^2 - 6x + 1\) and \(g'(x) = 2x + 2\)
Step 2 :Apply L'Hôpital's rule: \(\lim_{x \rightarrow 2} \frac{f'(x)}{g'(x)} = \lim_{x \rightarrow 2} \frac{3x^2 - 6x + 1}{2x + 2}\)
Step 3 :Calculate the limit: \(\lim_{x \rightarrow 2} \frac{3x^2 - 6x + 1}{2x + 2} = \frac{1}{6}\)
Step 4 :\(\boxed{\lim_{x \rightarrow 2} \frac{x^3 - 3x^2 + x + 2}{x^2 + 2x - 8} = \frac{1}{6}}\)