To convert the given general form of the equation of the ellipse to standard form, here are the steps:
1. Group the terms with the same variable. Transfer the constant term on the other side of the equation.
[tex](4x^2-32x)+(9y^2+36y)=-64[/tex]2. Extract a common factor on each group.
[tex]4(x^2-8x)+9(y^2+4y)=-64[/tex]3. Apply completing the square method in each group in parenthesis to make it a perfect square trinomial. Divide the middle term by 2 and square it. The result will be the third term in each group.
[tex]4(x^2-8x+16)+9(y^2+4y+4)=-64[/tex]4. Multiply the constant term in each group to each respective factor and add the results to the other side of the equation.
[tex]\begin{gathered} 4\times16=64 \\ 9\times4=36 \\ 4(x^2-8x+16)+9(y^2+4y+4)=-64+64+36 \\ 4(x^2-8x+16)+9(y^2+4y+4)=36 \end{gathered}[/tex]5. Rewrite each group of trinomial to binomial using its factors.
[tex]4(x-4)^2+9(y+2)^2=36[/tex]6. Lastly, divide both sides of the equation by the constant term 36.
[tex]\begin{gathered} \frac{4(x-4)^2}{36}+\frac{9(y+2)^2}{36}=\frac{36}{36} \\ \frac{(x-4)^2}{9}+\frac{(y+2)^2}{4}=1 \end{gathered}[/tex]Hence, the equation of the ellipse in standard form is:
[tex]\frac{(x-4)^2}{9}+\frac{(y+2)^2}{4}=1[/tex]