First, subtract #color(red)(4)# and add #color(blue)(x^2)# to each side of the equation to isolate the #y# term while keeping the equation balanced:
#4 - color(red)(4) - x^2 + color(blue)(x^2) + 16y^2 = 0 + color(blue)(x^2) - color(red)(4)#
#0 - 0 + 16y^2 = x^2 - 4#
#16y^2 = x^2 - 4#
Next, use this rule of quadratics to factor the right side of the equation:
#(color(red)(x) + color(blue)(y))(color(red)(x) - color(blue)(y)) = color(red)(x)^2 - color(blue)(y)^2#
#16y^2 = color(red)(x)^2 - color(blue)(4)#
#16y^2 = color(red)(x)^2 - color(blue)(2)^2#
#16y^2 = (color(red)(x) + color(blue)(2))(color(red)(x) - color(blue)(2))#
If you are required to continue and solve for #y#, take the square root of each side of the equation giving:
#sqrt(16y^2) = +-sqrt((color(red)(x) + color(blue)(2))(color(red)(x) - color(blue)(2)))#
#4y = +-sqrt((color(red)(x) + color(blue)(2))(color(red)(x) - color(blue)(2)))#
Then, divide each side of the equation by #color(green)(4)# to solve for #y#:
#(4y)/color(green)(4) = +-sqrt((color(red)(x) + color(blue)(2))(color(red)(x) - color(blue)(2)))/color(green)(4)#
#y = +-sqrt((color(red)(x) + color(blue)(2))(color(red)(x) - color(blue)(2)))/color(green)(4)#
