Sure, let's solve this problem step by step:
We need to calculate the difference between the two polynomials:
[tex]\[
P_1 = x^4 + x^3 + x^2 + x
\][/tex]
and
[tex]\[
P_2 = x^4 - x^3 + x^2 - x
\][/tex]
To find the difference \( P_1 - P_2 \), we will subtract \( P_2 \) from \( P_1 \):
1. Write down both polynomials:
[tex]\[
P_1 = x^4 + x^3 + x^2 + x
\][/tex]
[tex]\[
P_2 = x^4 - x^3 + x^2 - x
\][/tex]
2. Subtract \( P_2 \) from \( P_1 \):
[tex]\[
P_1 - P_2 = (x^4 + x^3 + x^2 + x) - (x^4 - x^3 + x^2 - x)
\][/tex]
3. Distribute the subtraction (subtract each term of \( P_2 \) from the corresponding term in \( P_1 \)):
[tex]\[
P_1 - P_2 = x^4 + x^3 + x^2 + x - x^4 + x^3 - x^2 + x
\][/tex]
4. Combine the like terms:
[tex]\[
= (x^4 - x^4) + (x^3 + x^3) + (x^2 - x^2) + (x + x)
\][/tex]
[tex]\[
= 0 + 2x^3 + 0 + 2x
\][/tex]
5. Simplify the expression to get the final result:
[tex]\[
= 2x^3 + 2x
\][/tex]
So, the difference of the polynomials \( \left(x^4 + x^3 + x^2 + x\right) - \left(x^4 - x^3 + x^2 - x\right) \) is:
[tex]\[
2x^3 + 2x
\][/tex]
Therefore, the correct answer is:
[tex]\[
2x^3 + 2x
\][/tex]
