how do i figure where this equation passes through the orgin
The tangent to the curve y = 1/x^2+8x+16 passing through the origin is very different to y = 1/x^2+8x+16 passing through the origin. Even though y = 1/x^2+8x+16 is completely ambiguous (is it 1/(x^2+8x+16) or (1/x^2) + 8x + 16 .....?) none of the possible interpretations of the equation of your curve pass through the origin.
Please be clearer what the equation of the curve is.