To combine like terms in the expression:

$y^3 - y^2 + 4y^3 + 7y^2 - 3x^3 + y^2 + 4y^3$

Let's first group the like terms:

• $y^3$-terms: $y^3 + 4y^3 + 4y^3$
• $y^2$-terms: $-y^2 + 7y^2 + y^2$
• $x^3$-term: $-3x^3$

Now, let's simplify each group:

• $y^3$-terms: $y^3 + 4y^3 + 4y^3 = 9y^3$
• $y^2$-terms: $-y^2 + 7y^2 + y^2 = 7y^2$
• $x^3$-term: $-3x^3$

So, the simplified expression is:

$9y^3 + 7y^2 - 3x^3$

Would you like further explanation or details?

Here are 5 related questions to explore:

1. How do you factor expressions like $9y^3 + 7y^2 - 3x^3$?
2. What are like terms in algebraic expressions?
3. How do you identify terms to combine in a polynomial?
4. How do coefficients affect combining like terms?
5. What are the differences between $x^3$ and $y^3$ in an algebraic context?

Tip: Always make sure to look for the same powers of variables when combining like terms in polynomials!