Question 737612
One change in sign so expect one positive root, according to Descartes Rule of Signs.  


(-x)^4+6(-x)^3+7(-x)^2-6(-x)-8
x^4-6x^3+7x^2+6x-8
That is three changes in sign, so expect three negative roots according to Rule of Signs.


Possible roots to check using Rational Roots Theorem and synthetic division would be -1, -2, -4, -8, 1, 2, 4, 8.


Best to check for the negative roots first because we expect MORE of them.


I will omit showing the actual synthetic divisions, but here is what I checked and what found:
Check -2: Yes, root.  Zero remainder. Quotient {{{x^3+4x^2-x-4}}}.
Check -4: Yes, root.  Zero remainder. Quotient {{{x^2-1 = (x-1)(x+1)}}}.
And from that last quotient, we see that roots are +1 and -1.


Summary of Roots found:
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-4, -2, -1, and +1
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