The problem requires finding the sum of two polynomials $f(x)$ and $g(x)$.

Given:

• $f(x) = x^3 + x^2 - 4$
• $g(x) = x^3 - 2x^2 + x + 2$

We need to find $f(x) + g(x)$.

Step-by-Step Solution

1. Add the corresponding terms of the two polynomials: $f(x) + g(x) = (x^3 + x^2 - 4) + (x^3 - 2x^2 + x + 2)$

2. Combine like terms:

   • Cubic terms: $x^3 + x^3 = 2x^3$
   • Quadratic terms: $x^2 - 2x^2 = -x^2$
   • Linear terms: $x = x$ (only one linear term from $g(x)$)
   • Constant terms: $-4 + 2 = -2$

3. Combine these terms to form the new polynomial: $f(x) + g(x) = 2x^3 - x^2 + x - 2$

Thus, the correct answer is: $\text{d. } 2x^3 - x^2 + x - 2$

Would you like more details, or do you have any questions?

Here are 5 related questions to expand your understanding:

1. What is the process of subtracting one polynomial from another?
2. How do you multiply two polynomials?
3. What are the rules for combining like terms in polynomial addition?
4. How would you find the product of $f(x)$ and $g(x)$ instead of their sum?
5. Can you represent the sum of two polynomials graphically?

Tip: When adding or subtracting polynomials, make sure to align and combine like terms (terms with the same power of $x$).