The polynomial p(x) = 2x3+10x2+10x+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.