$${x^7} + 3{x^5} - 13{x^3} - 15x = 0$$

$$x({x^6} + 3{x^4} - 13{x^2} - 15) = 0$$

$$x = 0 = {\alpha _7}$$

Let $${x^2} = t$$

$${t^3} + 3{t^2} - 13t - 15 = 0$$

$$(t + 1)(t + 5)(t - 3) = 0$$

$$t = {x^2} = - 1, - 5,3$$

$$x\, = \, \pm \,i, \pm \,\sqrt 5 i, \pm \,\sqrt 3 $$

$${\alpha _1},{\alpha _2} = \pm \,\sqrt 5 i,{\alpha _3},{\alpha _4} = \pm \,\sqrt 3 ,{\alpha _5},{\alpha _6} = \pm \,i$$

$${\alpha _1}{\alpha _2} - {\alpha _3}{\alpha _4} + {\alpha _5}{\alpha _6} = 5 + 3 + 1 = 9$$