How would I put $x^2 + 4x + 25y^2 - 50y = -4$ into the equation for an ellipse?

Question

The equation is $x^2 + 4x + 25y^2 - 50y = -4$. How would I put this into the equation for an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$?

Answer 1

Notice, make the perfect squares as follows $$x^2+4x+25y^2-50y=-4$$ $$(x^2+4x+4)-4+25(y^2-2y+1)-25=-4$$ $$(x+2)^2+25(y-1)^2=29-4=25$$ $$\frac{(x+2)^2}{25}+\frac{25(y-1)^2}{25}=1$$ $$\frac{(x+2)^2}{5^2}+\frac{(y-1)^2}{1^2}=1$$ you may now compare above equation with $\frac{X^2}{a^2}+\frac{Y^2}{b^2}=1$