Solve the expression (2x² + 7x + 33) / (x³ + x² - 1/x) into partial fractions.

Find two numbers that multiply to 12 and add to 7: 3 and 4.

$$n^{2} + 7n + 12 = (n + 3)(n + 4)$$

Substitute $n = -3$.

$$27 = a(-3 + 4) + b(-3 + 3)$$ $$27 = a(1) + b(0)$$ $$a = 27$$

Substitute $n = -4$.

$$27 = a(-4 + 4) + b(-4 + 3)$$ $$27 = a(0) + b(-1)$$ $$b = -27$$

$$\frac{27}{n + 3} + \frac{-27}{n + 4} = \frac{27}{n + 3} - \frac{27}{n + 4}$$

Final answer:
\dfrac{27}{n+3} - \dfrac{27}{n+4}