Beginning Algebra With Applications, Chapter 7, 7.3, Section 7.3, Problem 154

Simplify: $(x + 1)\left( -x^6 + x^5 - x^4 + x^3 - x^2 + x - 1 \right)$

$
\begin{equation}
\begin{aligned}
&= x(-x^6) + x(x^5) - x(x^4) + x(x^3) - x(x^2) + x(x) -x -x^6 + x^5 - x^4 + x^3 - x^2 + x - 1
&& \text{Apply Distributive Property}\\
\\
&= -x^7 + x^6 - x^5 + x^4 - x ^3 + x^2 - x - x^6 + x^5 - x^4 + x^3 - x^2 + x - 1
&& \text{Evaluate the expression}\\
\\
&= -x^7 + x^6 - x^6 - x^5 + x^5 + x^4 - x^4 - x^3 + x^3 + x^2 - x^2 - x + x - 1
&& \text{Group terms}\\
\\
&= -x^7 - 1
&& \text{Combine like terms}
\end{aligned}
\end{equation}
$