To solve the given problem graphically, let's break it down step by step:

1. Given Equations:

   • The quadratic function provided is $y = x^2 + 8x - 3$.
   • The equation you want to solve graphically is $x^2 + 10x - 12 = 0$.

2. Line Equation Form: To solve this graphically, we rewrite $x^2 + 10x - 12 = 0$ as $x^2 + 8x - 3 = -2x - 9$. This means we want the line $y = -2x - 9$ to intersect the parabola $y = x^2 + 8x - 3$.

3. Required Line: The line equation is $y = -2x - 9$.

Explanation:

By drawing the parabola $y = x^2 + 8x - 3$ and the line $y = -2x - 9$ on the same graph, their points of intersection will provide the solutions to the equation $x^2 + 10x - 12 = 0$.

Final Answer:

The line to draw is: $y = -2x - 9$

Let me know if you need further details or visualizations for this solution.

Here are 5 follow-up questions for deeper understanding:

1. How do you find the intersection points between a parabola and a line graphically?
2. What is the geometric interpretation of the roots of a quadratic equation?
3. Could we solve $x^2 + 10x - 12 = 0$ algebraically, and how would that compare to the graphical solution?
4. How does the slope $m$ of a line influence its intersection with a given curve?
5. How would you adjust the method if a cubic curve were involved instead of a quadratic?

Tip: Always ensure both the curve and the line are plotted accurately to find the solutions graphically!