SOLUTION: Find the Standard equation of hyperbola, center, foci, vertices at asymptotes of the function 4x²-5y²+32x+30y=1. Thank you!

4x^2 - 5y^2 + 32x + 30y = 1  ====>  Complete the squares for x- and y-terms separately  ====>  


(4x^2 + 32x) + (-5y^2 + 30y) = 1


4(x^2 + 8x) - 5*(y^2 - 6y) = 1


4*(x^2 + 8x + 16) - 5*(y^2 - 6y + 9) = 1 + 4*16 - 5*9


4*(x+4)^2 - 5*(y-3)^2 = 20.     <<<---=== Divide by 20 both sides


 -  = 1


 -  = 1


It is the standard form equation for a hyperbola.


The hyperbola has the center at  (x,y) = (-4,3).


Real axis is parallel to x-axis; imaginary axis is parallel to y-axis.


Real semi-axis is  units long;  Imaginary axis is 2 units long.


Vertices are at y= 3:  x =   and  x= .


The foci are  x=  = -4 - 3 = -7  and  x=  = -4 + 3 = -1.