Answer:
Graphically.
Explanation:
We have the equations:
(x + 6)^2 = -12*(y + 2)
(x^2/81) + (y^2/100) = 1
First, we can see that the second one is an ellipse, and we can write it as:
(x/9)^2 + (y/10)^2 = 1
This adds some complexity to our problem.
The first equation is just a quadratic equation, that we can write in standard form as:
y = (x^2 + 12*x + 36)/(-12) - 2
y = (-1/12)*x^2 - x - 5
Then our two equations are:
(x/9)^2 + (y/10)^2 = 1
y = (-1/12)*x^2 - x - 5
This is actually really hard to solve analytically, we may end with a quartic equation or something like that (this happens for the nature of the ellipse equation)
So now that we have two rather simple equations that we know how to graph (or that we can just input the equations in a program and graph them) we can find the intersections visually.
You can see the graph of this below:
Of course, this only works to find estimations for the intersections, where one intersection is (3.35, -9.28) and the other is at (-8.69, -2.6)