1. **State the problem:** We want to simplify the expression $$\frac{2x^2 + 7x + 33}{x^3 + x^2 - 11x}$$\n\n2. **Factor the denominator:** Look for common factors first. The denominator is $$x^3 + x^2 - 11x$$. Factor out $$x$$:\n$$x^3 + x^2 - 11x = x(x^2 + x - 11)$$\n\n3. **Check if the quadratic $$x^2 + x - 11$$ can be factored:** The discriminant is $$\Delta = 1^2 - 4 \times 1 \times (-11) = 1 + 44 = 45$$, which is positive but not a perfect square, so it does not factor nicely over the integers.\n\n4. **Try polynomial division or partial fraction decomposition:** Since numerator degree 2 is less than denominator degree 3, cannot divide further.\n\n5. **Rewrite the expression:** $$\frac{2x^2 + 7x + 33}{x(x^2 + x - 11)}$$\n\n6. **Since no obvious common factors, this is the simplest form unless complex factors are allowed.**\n\n**Final answer:** $$\frac{2x^2 + 7x + 33}{x(x^2 + x - 11)}$$