The polynomial *p*(*x*) = 2*x*^{3}+10*x*^{2}+10*x*+2
clearly has a root at *x* = -1, leading to the factorization

.

The roots of the quadratic factor can be found by completing the square
or with the quadratic formula. They turn out to be

.

Now we know that -1 = -tan(45). We can use the half-angle formula for
tangent to see that the remaining roots are -tan(15) and -tan(75).
The half-angle formula says

.

Taking A = 30 leads to

,

and with A = 150 we find

.

This shows that the quadratic factor of the polynomial has -tan(15) and -tan(75) as roots.