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

Let's examine the given equation:

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

We'll start by opening the second and third pairs of parentheses from the left (marked with an underline below) where in the left side we note that we can use the difference of squares formula:

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

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

$(x+1)\underline{(x-1)(x+1)}=x^2+x^3 \\ \downarrow\\ (x+1)\underline{\textcolor{blue}{(}x^2-1^2\textcolor{blue}{)}}=x^2+x^3\\ (x+1)\underline{\textcolor{blue}{(}x^2-1\textcolor{blue}{)}}=x^2+x^3\\$We'll continue and use the expanded distributive law again and open the parentheses on the left side, then we'll move terms and combine like terms:

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

where in the final step we simply solved the first-degree equation that we got,

therefore the correct answer is answer A.