Although most textbooks use x' and y', for convenience I
will use capital X instead of x' and capital Y instead of y'.

The angle of rotation  of the graph 

Ax² + Bxy + Cy² + Dx + Ey + F = 0

necessary to transform it into an equation in X and Y which contains
no term in XY

is calculated by  or  if A = C

Your equation is

x² + 8xy + y² - 15 = 0.

In this case A-1, B=8, C=1, D=0, E=0, F = -15

Since A=C=1 the required angle of rotation is 

The substitutions to make are 




Since , the substitutions are




or




or





x² + 8xy + y² - 15 = 0 becomes



Multiply through by 2









Divide through by 30 to get 1 on the right side:





Edwin