solve the equation in real numbers system x^4+8x^3+6x^2-5x+14=0``please show steps so i can learn how to do this on my own

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The equation x^4+8x^3+6x^2-5x+14=0 has to be solved.

x^4+8x^3+6x^2-5x+14=0

=> x^4+7x^3+x^3 + 7x^2 - x^2 - 7x + 2x + 14=0

=> x^3(x + 7)+x^2(x + 7) - x(x + 7) + 2(x + 7) = 0

=> (x + 7)(x^3+x^2- x+ 2) = 0

=> (x + 7)(x^3+ 2x^2 - x^2 - 2x + x + 2) = 0

=> (x + 7)(x^2(x + 2) - x(x + 2) + 1(x + 2)) = 0

=> (x + 7)(x + 2)(x^2 - x + 1) = 0

x + 7 = 0 => x = -7

x + 2 = 0 => x = -2

x^2 - x + 1 = 0

=> x = `(1 + sqrt(1 - 4))/2` and x = `(1 - sqrt(1 - 4))/2`

=> x = `(1+ isqrt3)/2` and x = `(1 - i*sqrt 3)/2`

The roots of the equation are `(-7, -2, (1+ isqrt3)/2,(1 - i*sqrt 3)/2)`

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