Question 45475
Descartes' Rule of Signs 
Given a polynomial such as:

x^4 + 7x^3 - 4x^2 - x - 7 

Is it possible to say anything about how many positive real roots it has, just by looking at it? 
Here's a striking theorem due to Descartes in 1637, often known as "Descartes' rule of signs": The number of positive real roots of a polynomial is bounded by the number of changes of sign in its coefficients. Gauss later showed that the number of positive real roots, counted with multiplicity, is of the same parity as the number of changes of sign. 

Thus for the polynomial above, there is at most one positive root, and therefore exactly one. 

In fact, an easy corollary of Descartes' rule is that the number of negative real roots of a polynomial f(x) is determined by the number of changes of sign in the coefficients of f(-x). So in the example above, the number of negative real roots must be either 1 or 3. 
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Your problem: 
g(x)=x^4+x^3+2x^2-3x-1

#of changes of signs = 1
So, #of positive real zeroes =1

g(-x)=x^4-x^3+2x^2+3x-1
Its # of changes of signs = 3
So, # of negative real zeroes = 1 or 3

Cheers,
Stan H.