SAT Question a4f61d75: Nonlinear Equations In One Variable And Systems Of Equations In Two Variables answer and explanation with Desmos solution

Since the given equation has two integer solutions, the expression on the left-hand side of this equation can be factored as $(x+c)(x+d)$, where $c$ and $d$ are also integers. The product of $c$ and $d$ must equal the constant term of the original quadratic expression, which is $12$. Additionally, the sum of $c$ and $d$ must be a negative number since it's given that $a>0$, but the sign preceding $a$ in the given equation is negative. The possible pairs of values for $c$ and $d$ that satisfy both of these conditions are $-4$ and $-3$, $-6$ and $-2$, and $-12$ and $-1$. Since the value of $-a$ is the sum of $c$ and $d$, the possible values of $-a$ are $-4+(-3)=-7$, $-6+(-2)=-8$, and $-12+(-1)=-13$. It follows that the possible values of $a$ are $7$, $8$, and $13$. Note that 7, 8, and 13 are examples of ways to enter a correct answer.