To find the difference between the two polynomials [tex]\( \left(x^4 + x^3 + x^2 + x\right) \)[/tex] and [tex]\( \left(x^4 - x^3 + x^2 - x\right) \)[/tex], we need to perform polynomial subtraction:
Step 1: Write down the given polynomials.
The first polynomial is:
[tex]\[ x^4 + x^3 + x^2 + x \][/tex]
The second polynomial is:
[tex]\[ x^4 - x^3 + x^2 - x \][/tex]
Step 2: Align the terms by their degrees and then subtract the second polynomial from the first.
[tex]\[
\begin{array}{r}
\left(x^4 + x^3 + x^2 + x \right) \\
-\left(x^4 - x^3 + x^2 - x \right) \\
\end{array}
\][/tex]
Step 3: Perform the subtraction term by term:
[tex]\[
\begin{array}{rl}
x^4 \phantom{{}-{}} & - x^4 = 0 \\
x^3 \phantom{{}+{}} & - (- x^3) = x^3 + x^3 = 2x^3 \\
x^2 \phantom{{}-{}} & - x^2 = 0 \\
x \phantom{{}+{}} & - (- x) = x + x = 2x \\
\end{array}
\][/tex]
Step 4: Combine the results:
[tex]\[
0x^4 + 2x^3 + 0x^2 + 2x
\][/tex]
So the equation simplifies to:
[tex]\[
2x^3 + 2x
\][/tex]
Therefore, the difference of the polynomials is:
[tex]\[ 2x^3 + 2x \][/tex]
Among the provided options, the correct one is:
[tex]\[ 2x^3 + 2x \][/tex]
