To solve the expression $(x^4 + 2x^3 + x^2 + 4) - (-x^2 + x^3 - x^4 - 5x + 7)$, follow these steps:

Step 1: Distribute the negative sign

First, distribute the negative sign across the terms inside the second set of parentheses:

$(x^4 + 2x^3 + x^2 + 4) - (-x^2 + x^3 - x^4 - 5x + 7)$

$= (x^4 + 2x^3 + x^2 + 4) + (x^2 - x^3 + x^4 + 5x - 7)$

Step 2: Combine like terms

Now, combine the like terms:

$x^4 + x^4 + 2x^3 - x^3 + x^2 + x^2 + 5x + 4 - 7$

$= 2x^4 + x^3 + 2x^2 + 5x - 3$

So, the simplified expression is:

$2x^4 + x^3 + 2x^2 + 5x - 3$

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. How would you solve a polynomial equation like $2x^4 + x^3 + 2x^2 + 5x - 3 = 0$?
2. Can you factor the expression $2x^4 + x^3 + 2x^2 + 5x - 3$?
3. What are the roots of a polynomial expression, and how can they be found?
4. How do you graph the polynomial $2x^4 + x^3 + 2x^2 + 5x - 3$?
5. How does the degree of a polynomial affect its graph?

Tip: Always distribute negative signs carefully, as missing one can lead to incorrect results.