Its a nested quadratic :P

To see that more clearly First write

So

4(x – 1/x)² + 8(x + 1/x) – 29 = 4((x + 1/x)² - 4) + 8(x + 1/x) – 29 = 4(x + 1/x)² + 8(x + 1/x) – 45 = 0

Now the last equation is a quadratic in (x+1/x). So solve it. Once you get a couple of answers for that solve the equation "x+1/x = previous answer", which is another quadratic.