1. **Stating the problem:** Solve the equation $$\sqrt{x^3 + x^2 - 1} + \sqrt{x^3 + x^2 + 2} = 3.$$\n\n2. **Understanding the problem:** We have two square root expressions added together equal to 3. Our goal is to find all real values of $x$ that satisfy this equation.\n\n3. **Isolate and simplify:** Let $A = \sqrt{x^3 + x^2 - 1}$ and $B = \sqrt{x^3 + x^2 + 2}$. The equation is $A + B = 3$.\n\n4. **Square both sides:** \n$$ (A + B)^2 = 3^2 \Rightarrow A^2 + 2AB + B^2 = 9. $$\n\n5. **Substitute back:** \n$$ (x^3 + x^2 - 1) + 2\sqrt{(x^3 + x^2 - 1)(x^3 + x^2 + 2)} + (x^3 + x^2 + 2) = 9. $$\n\n6. **Combine like terms:** \n$$ 2x^3 + 2x^2 + 1 + 2\sqrt{(x^3 + x^2 - 1)(x^3 + x^2 + 2)} = 9. $$\n\n7. **Isolate the square root term:** \n$$ 2\sqrt{(x^3 + x^2 - 1)(x^3 + x^2 + 2)} = 9 - (2x^3 + 2x^2 + 1) = 8 - 2x^3 - 2x^2. $$\n\n8. **Divide both sides by 2:** \n$$ \sqrt{(x^3 + x^2 - 1)(x^3 + x^2 + 2)} = 4 - x^3 - x^2. $$\n\n9. **Square both sides again:** \n$$ (x^3 + x^2 - 1)(x^3 + x^2 + 2) = (4 - x^3 - x^2)^2. $$\n\n10. **Expand left side:** \n$$ (x^3 + x^2)^2 + 2(x^3 + x^2) - 1(x^3 + x^2) - 2 = (x^3 + x^2)^2 + (2 - 1)(x^3 + x^2) - 2 = (x^3 + x^2)^2 + (x^3 + x^2) - 2. $$\n\n11. **Expand right side:** \n$$ (4 - x^3 - x^2)^2 = 16 - 8(x^3 + x^2) + (x^3 + x^2)^2. $$\n\n12. **Set equation:** \n$$ (x^3 + x^2)^2 + (x^3 + x^2) - 2 = 16 - 8(x^3 + x^2) + (x^3 + x^2)^2. $$\n\n13. **Subtract $(x^3 + x^2)^2$ from both sides:** \n$$ (x^3 + x^2) - 2 = 16 - 8(x^3 + x^2). $$\n\n14. **Bring all terms to one side:** \n$$ (x^3 + x^2) - 2 - 16 + 8(x^3 + x^2) = 0 \Rightarrow 9(x^3 + x^2) - 18 = 0. $$\n\n15. **Simplify:** \n$$ 9(x^3 + x^2) = 18 \Rightarrow x^3 + x^2 = 2. $$\n\n16. **Rewrite:** \n$$ x^2(x + 1) = 2. $$\n\n17. **Solve for $x$:** This is a cubic equation $x^3 + x^2 - 2 = 0$.\n\n18. **Try rational roots:** Possible roots are factors of 2: $\pm1, \pm2$.\n\n- For $x=1$: $1 + 1 - 2 = 0$ ✓ root found.\n- For $x=-2$: $-8 + 4 - 2 = -6 \neq 0$.\n\n19. **Factor out $(x-1)$:** \n$$ x^3 + x^2 - 2 = (x - 1)(x^2 + 2x + 2). $$\n\n20. **Solve quadratic:** \n$$ x^2 + 2x + 2 = 0. $$\nDiscriminant: $\Delta = 4 - 8 = -4 < 0$, no real roots.\n\n21. **Real solution:** \n$$ x = 1. $$\n\n22. **Check solution in original equation:** \n$$ \sqrt{1 + 1 - 1} + \sqrt{1 + 1 + 2} = \sqrt{1} + \sqrt{4} = 1 + 2 = 3. $$\nCorrect.\n\n**Final answer:** \n$$ \boxed{x = 1}. $$