The polynomial p(x) = 2x3+10x2+10x+2 clearly has a root at x = -1, leading to the factorization

p(x)=2(x+1)(x^2+4x+1) .

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

x = 2 +/- sqrt(3) .

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

tan(A/2) = (1-cos A)/(sin A) .

Taking A = 30 leads to

tan(15) = 2-sqrt(3) ,

and with A = 150 we find

tan(75) = 2+sqrt(3) .

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