Solve (x+1)(x-1)(x+1) = x^3 + x^2: Cubic Equation with Repeated Factor

Solve the following problem:

$(x+1)(x-1)(x+1)=x^2+x^3$

Let's examine the given equation:

$(x+1)(x-1)(x+1)=x^2+x^3$

Begin by opening the second and third pairs of parentheses from the left (marked with an underline below) On the left side we can apply the difference of squares formula:

$(a+b)(a-b)=a^2-b^2$,

Place the result inside of new parentheses (since the resulting expression in its entirety is multiplied by an expression that is enclosed by these parentheses) then we'll proceed to simplify the expression in the resulting parentheses:

$(x+1)\underline{(x-1)(x+1)}=x^2+x^3 \\ \downarrow\\ (x+1)\underline{\textcolor{blue}{(}x^2-1^2\textcolor{blue}{)}}=x^2+x^3\\ (x+1)\underline{\textcolor{blue}{(}x^2-1\textcolor{blue}{)}}=x^2+x^3\\$Continue to use the expanded distributive law again and open the parentheses on the left side. Proceed to move and combine like terms:

$(x+1)(x^2-1)=x^2+x^3\\ \downarrow\\ x^3-x+x^2-1=x^2+x^3\\ -x =1\hspace{6pt}\text{/}\cdot(-1)\\ \downarrow\\ \boxed{x=-1}$

In the final step solve the first-degree equation that we obtained,

Therefore the correct answer is answer A.