SAT Question e9349667: Nonlinear Equations In One Variable And Systems Of Equations In Two Variables answer and explanation with Desmos solution

Choice D is correct. The system of equations can be solved using the substitution method. Solving the second equation for $y$ gives $y=-x-1$. Substituting the expression $-x-1$ for $y$ into the first equation gives $-x-1=x^2+2x+1$. Adding $x+1$ to both sides of the equation yields $x^2+3x+2=0$. The left-hand side of the equation can be factored by finding two numbers whose sum is $3$ and whose product is $2$, which gives $(x+2)(x+1)=0$. Setting each factor equal to $0$ yields $x+2=0$ and $x+1=0$, and solving for $x$ yields $x=-2$ or $x=-1$. These values of $x$ can be substituted for $x$ in the equation $y=-x-1$ to find the corresponding $y$-values: $y=-(-2)-1=2-1=1$ and $y=-(-1)-1=1-1=0$. It follows that $(-2,1)$ and $(-1,0)$ are the solutions to the given system of equations. Therefore, $(x_1,y_1)=(-2,1)$, $(x_2,y_2)=(-1,0)$, and $y_1+y_2=1+0=1$.