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.

The query submitted by the user: (x^4 + 2x^3 + x^2 + 4) - (-x^2 + x^3 - x^4 - 5x + 7)

This page explains how to simplify the polynomial expression (x^4 + 2x^3 + x^2 + 4) - (-x^2 + x^3 - x^4 - 5x + 7). Follow a detailed step-by-step solution, covering key concepts like distributing negative signs and combining like terms. Suitable for students in Grades 9-12.

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