Let's simplify the given expression step by step:

The expression is:

$7x^2 - 4x^2y - x^2 + 8x^2y$

1. Combine the like terms involving $x^2$: $7x^2 - x^2 = (7 - 1)x^2 = 6x^2$

2. Now combine the terms involving $x^2y$: $-4x^2y + 8x^2y = (8 - 4)x^2y = 4x^2y$

3. Finally, combine the results: $6x^2 + 4x^2y$

So, the simplified expression is:

$6x^2 + 4x^2y$

Would you like a detailed breakdown or have any questions?

Here are 5 related questions to explore further:

1. How would you factorize the expression $6x^2 + 4x^2y$?
2. How can you identify like terms in algebraic expressions?
3. What is the difference between a monomial and a binomial?
4. How can expressions with multiple variables, like $x^2y$, be simplified further?
5. Can you apply this method to simplify $9x^2 - 5x^2y + 3x^2y - 2x^2$?

Tip: Always look for like terms (same variables raised to the same power) when simplifying algebraic expressions.