2x^3 + 8x^2 - 120x + k = 0 divide through by 2
x^3 + 4x^2 - 60x + k/2 = 0 (1)
Let the two equal roots = m and let the other root be n
So we have
(x - m)^2 (x - n) = 0
(x^2 - 2xm + m^2) ( x - n) = 0
x^3 -2mx^2 + m^2x - x^2n + 2mnx -m^2n = 0
x^3 - (2m + n)x^2 + (m^2+2mn)x -m^2n = 0 (2)
Equating coefficients in (1) and (2) we have the following system
-(2m + n) = 4 ⇒ 2m + n = -4 ⇒ n = -4 - 2m ( 3)
(m^2 +2mn) = -60 (4)
-m^2n = k/2 ⇒ -2m^2n = k (5)
Sub (3) into (4) for n and we have that
m^2 + 2m (-4 -2m) = -60
m^2 - 8m - 4m^2 = -60
-3m^2 -8m + 60 = 0
3m^2 + 8m - 60 = 0 factor
(3m - 10) ( m + 6) = 0
Set each factor to 0 and solve for m and we have that m =10/3 or m = -6
If m = -6 then n = -4 - 2(-6) = 8 and using (5) gives -2(-6)^2 * 8 = -576 = k [reject]
If m = 10/3 then n = -4 - 2(10/3) = -32/3 and (5) gives -2 (10/3)^2* (-32/3) = 6400/27 = k