Let's say P(x) = x^4 + 7x^2 + 10 and let's say X = x^2. So we're now studying P(X) = X^2 + 7X + 10, which is a classic trinomial we can solve easily.
In order to use the quadratic formula, we first need to calculate Delta = b^2 - 4ac. Here, Delta = 7^2 -4*10 = 9. So this polynomial has 2 real roots.
By the quadratic formula, the roots are given by (-b +- sqrtDelta)/2a.
X_1 = (-7 - 3)/2 = -10/2 = -5 and X_2 = (-7 + 3)/2 = -4/2 = -2.
So P(X) = (X+5)(X+2). We now switch back to the original variable, hence P(x) = (x^2 + 5)(x^2 + 2). If we want to factorise it completely, there is still some work to do with complex numbers.
I will only show you how to solve x^2 + 5 = 0 in CC because it's the exact same way to solve x^2 + 2 = 0.
In CC, x^2 + 5 = 0 iff x^2 = -5 iff x = +-isqrt5. So the complete factorization of P in CC is (x - isqrt5)(x + isqrt5)(x - isqrt2)(x + isqrt2).
