1. **State the problem:** Simplify the expression $$\frac{3x-6}{x^2} + x - 6$$. 2. **Rewrite the expression:** The expression is a sum of a fraction and two terms: $$\frac{3x-6}{x^2} + x - 6$$. 3. **Factor the numerator of the fraction:** $$3x - 6 = 3(x - 2)$$. 4. **Rewrite the fraction:** $$\frac{3(x-2)}{x^2}$$. 5. **Express the entire expression with a common denominator:** The common denominator is $$x^2$$, so rewrite $$x$$ and $$-6$$ as fractions: $$x = \frac{x^3}{x^2}$$ and $$-6 = \frac{-6x^2}{x^2}$$. 6. **Combine all terms over the common denominator:** $$\frac{3(x-2)}{x^2} + \frac{x^3}{x^2} - \frac{6x^2}{x^2} = \frac{3(x-2) + x^3 - 6x^2}{x^2}$$. 7. **Expand and simplify the numerator:** $$3(x-2) + x^3 - 6x^2 = 3x - 6 + x^3 - 6x^2$$. 8. **Rearrange terms in descending powers of $$x$$:** $$x^3 - 6x^2 + 3x - 6$$. 9. **Factor the numerator if possible:** Group terms: $$ (x^3 - 6x^2) + (3x - 6) = x^2(x - 6) + 3(x - 2)$$. No common factor between $$x^2(x-6)$$ and $$3(x-2)$$, so try factoring by grouping differently or use polynomial division. 10. **Try factoring the cubic polynomial:** Test possible roots using Rational Root Theorem: possible roots are $$\pm1, \pm2, \pm3, \pm6$$. 11. **Test $$x=2$$:** $$2^3 - 6(2)^2 + 3(2) - 6 = 8 - 24 + 6 - 6 = -16$$ (not zero). 12. **Test $$x=3$$:** $$3^3 - 6(3)^2 + 3(3) - 6 = 27 - 54 + 9 - 6 = -24$$ (not zero). 13. **Test $$x=1$$:** $$1 - 6 + 3 - 6 = -8$$ (not zero). 14. **Test $$x=-1$$:** $$-1 - 6 + (-3) - 6 = -16$$ (not zero). 15. **Test $$x=-2$$:** $$-8 - 24 - 6 - 6 = -44$$ (not zero). 16. **Test $$x=-3$$:** $$-27 - 54 - 9 - 6 = -96$$ (not zero). 17. **Test $$x=-6$$:** $$-216 - 216 - 18 - 6 = -456$$ (not zero). 18. Since no rational roots, the numerator does not factor nicely over rationals. 19. **Final simplified form:** $$\frac{x^3 - 6x^2 + 3x - 6}{x^2}$$. 20. **Optional: split the fraction:** $$\frac{x^3}{x^2} - \frac{6x^2}{x^2} + \frac{3x}{x^2} - \frac{6}{x^2} = x - 6 + \frac{3}{x} - \frac{6}{x^2}$$. **Answer:** $$x - 6 + \frac{3}{x} - \frac{6}{x^2}$$.