How to solve #x^4+2x^3-25x^2-26x+120=0# given that the product of two of its roots is 8?

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I need to understand how to solve bi-quadratic equations efficiently.

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Answer 1 · 2017-05-23T19:47:55

Tthe roots are, #-5, -3, 2 and 4.#

#"The Expression="x^4+2x^3-25x^2-26x+120,#

#=x^4+2x^3+x^2-26x^2-26x+120,......[because," completing the square]"#

#=(x^2+x)^2-26(x^2+x)+120,#

#=y^2-26y+120, where, y=x^2+x,#

#=ul(y^2-20y)-ul(6y+120),.........[20xx6=120, &, 20+6=26],#

#=y(y-20)-6(y-20),#

#=(y-20)(y-6),#

#=(x^2+x-20)(x^2+x-6),#

#=(x+5)(x-4)(x+3)(x-2).#

Hence, the roots are, #-5, -3, 2 and 4.#