Once you have $y$ written in terms of $x$, plug that expression into $x^2 - y = 12$ so that the second equation only involves $x$.

After substituting, you should get a quadratic equation in $x$. Rearrange it to equal $0$, factor if possible, and solve for $x$.

You will get two values of $x$ from the quadratic. Check which of these values appears in the answer choices.

Rewrite the system as $y = 8 - x$ and $y = x^2 - 12$ so each equation is in the form $y = \text{expression in } x$.

In Desmos, enter y = 8 - x on one line and y = x^2 - 12 on another. You should see a straight line and a parabola on the graph.

Use Desmos to find the intersection points of the line and the parabola (tap or click where they cross). Note the $x$-coordinates of these intersection points, and then see which of the answer choices matches one of those $x$-values.

From the linear equation

solve for $y$ in terms of $x$:

Now you can substitute this expression for $y$ into the second equation.

Take the second equation

and substitute $y = 8 - x$:

Now simplify this equation so it only has $x$.

Simplify the left side:

So the equation becomes

Move all terms to one side:

Now you have a quadratic equation in $x$.

Factor the quadratic

Look for two numbers that multiply to $-20$ and add to $1$. Those numbers are $5$ and $-4$, so the equation factors as

We will solve for $x$ and select the matching answer choice in the next step.

Set each factor equal to zero and solve:

Compare the possible $x$-values $-5$ and $4$ with the options:

• $-2$ is not one of the solutions.
• $1$ is not one of the solutions.
• $4$ is one of the solutions.
• $6$ is not one of the solutions.

Therefore, the correct answer choice is $4$.