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

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

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