<s>[INST] Finding the Equation of the Ellipse By Completing the Square I have the following equation of an ellipse:
$$5x^2+9y^2+40x=100$$
I need to put it in the form: 
$$\frac{(x-h)^2}{m^2} + \frac{(y-k)^2}{n^2} =1 $$ 
I was trying to complete the square with the coefficients that have an $x$ variable. 
$$5x^2 + 40x + 9y^2 = 100$$
$$5(x^2 + 8x) + 9y^2 = 100$$
$$5(x^2 + 8x + 16) + 9y^2 = 100 + 16$$
$$5(x+4)^2 + 9y^2 = 116$$
I then divided both sides of the equation by $116$. 
$$\frac{5(x + 4)^2}{116} + \frac{9y^2}{116} = 1$$
However, when I graph the equation I do not get the same ellipse that was represented with the original equation of $5x^2 + 9y^2 + 40x = 100$. From the graph, it seems like they are two similar ellipses. Where am I making the mistake? Any help will be appreciated. 
 [/INST] We have
$$5x^2+9y^2+40x=100\iff 5(x+4)^2+9y^2=180\iff (x+4)^2+\frac{y^2}{\frac59}=36\\\iff\frac{(x+4)^2}{36}+\frac{y^2}{20}=1$$
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