To add the two given polynomials, we need to combine like terms. Let's break it down step-by-step:
We start with the given polynomials:
[tex]\[
(2x^4 + x^3 - 4x^2 + 3) + (4x^4 - 2x^3 - x^2 + 10x + 2)
\][/tex]
Step 1: Identify and combine like terms.
### \(x^4\) terms:
The \(x^4\) terms are \(2x^4\) and \(4x^4\):
[tex]\[
2x^4 + 4x^4 = 6x^4
\][/tex]
### \(x^3\) terms:
The \(x^3\) terms are \(x^3\) and \(-2x^3\):
[tex]\[
x^3 - 2x^3 = -x^3
\][/tex]
### \(x^2\) terms:
The \(x^2\) terms are \(-4x^2\) and \(-x^2\):
[tex]\[
-4x^2 - x^2 = -5x^2
\][/tex]
### \(x\) terms:
The \(x\) term is \(10x\):
[tex]\[
10x
\][/tex]
### Constant terms:
The constants are \(3\) and \(2\):
[tex]\[
3 + 2 = 5
\][/tex]
Step 2: Combine all the like terms to write the final polynomial in standard form:
[tex]\[
6x^4 - x^3 - 5x^2 + 10x + 5
\][/tex]
So, the sum of the polynomials \( \left(2 x^4 + x^3 - 4 x^2 + 3 \right) + \left(4 x^4 - 2 x^3 - x^2 + 10 x + 2 \right) \) is:
[tex]\[
6x^4 - x^3 - 5x^2 + 10x + 5
\][/tex]
