This is one possible way to solve this problem. For my purposes, I will use the symbols a, b, c, and d to represent the four roots.
From the original statement, we know that a + b = 0. This means that x = a and x = -a are roots of the equation. Furthermore, that means that f(a) = 0 and f(-a) = 0, since roots of equations are also places where the equation evaluates to a value of 0. If we evaluate a and -a into the original equation, we can derive a set of two equations which we can solve simultaneously:
f(a) = a^4 - 2a^3 + 4a^2 + 6a - 21 = 0
f(-a) = a^4 + 2a^3 + 4a^2 - 6a - 21 = 0
Since both are equal to 0, we can add each together to get a new equation which also equals 0:
f(a) + f(-a) = 0 =>
2a^4 + 8a^2 - 42 = 0
a^4 + 4a^2 - 21 = 0
This last equation we can further breakdown by considering the substitution w = a^2 and rewriting:
w^2 + 4w - 21 = 0
Solving this with the Quadratic Formula yields possible values for w:
w = (-4 +-sqrt(4^2-4*1*(-21)))/2 = (-4 +- sqrt(100))/2
w = (-4 +- 10)/2 = 3 or -7
Recalling that we had substituted w = a^2, we can finally work out the value(s) of a:
w = a^2
{(a^2 = 3, a = +-sqrt(3)), (a^2 = -7, "No solutions"):}
We have two of the roots found: x = sqrt(3), x = -sqrt(3). Noting that these represent the factors (x-sqrt(3))(x + sqrt(3)), we can now divide the original equation by these two factors (the work of which I will leave to the reader), and we reduce the equation down to the following:
x^2 - 2x + 7
This can also be solved using the Quadratic Formula:
x = (2 +- sqrt((-2)^2 - 4*1*7))/2 = (2 +- sqrt(-24))/2
= (2 +- 2isqrt(6))/2 = 1 +- isqrt(6)
Our final two roots are x = 1 + isqrt(6), x = 1 - isqrt(6).