1. **State the problem:** Simplify or analyze the rational expression $$\frac{3x^3 + 7x - 4}{(x^2 + 2)^2}$$. 2. **Understand the components:** The numerator is a cubic polynomial $$3x^3 + 7x - 4$$ and the denominator is the square of a quadratic polynomial $$(x^2 + 2)^2$$. 3. **Check for factorization:** Try to factor the numerator if possible. 4. **Factor numerator:** Use Rational Root Theorem to test possible roots for $3x^3 + 7x - 4$. 5. Test $x=1$: $$3(1)^3 + 7(1) - 4 = 3 + 7 - 4 = 6 \neq 0$$ 6. Test $x=-1$: $$3(-1)^3 + 7(-1) - 4 = -3 -7 -4 = -14 \neq 0$$ 7. Test $x=\frac{1}{3}$: $$3\left(\frac{1}{3}\right)^3 + 7\left(\frac{1}{3}\right) - 4 = 3\left(\frac{1}{27}\right) + \frac{7}{3} - 4 = \frac{1}{9} + \frac{7}{3} - 4 = \frac{1}{9} + \frac{21}{9} - \frac{36}{9} = -\frac{14}{9} \neq 0$$ 8. No obvious rational roots; numerator does not factor nicely over rationals. 9. **Conclusion:** The expression is already in simplest form since denominator is squared and numerator does not factor to cancel terms. 10. **Final answer:** $$\boxed{\frac{3x^3 + 7x - 4}{(x^2 + 2)^2}}$$