Question 1010745
The analysis you are using is called Descartes' rule of signs.

When you have determined the number of sign changes for {{{f(x)}}}, then you know the (maximum) possible number of positive zeros for your function.

You then have to evaluate {{{f(-x)}}} to get the (maximum) possible number of {{{negative}}} zeros.

Positive and negative possible zeros always decrease by two's. You can't go below none.

In your problem:
{{{g(x)=x^4+3x^3+7x^2-6x-13 }}}

there are {{{1}}} sign changes for {{{( + x)}}}, which means there is {{{1}}}  positive real zero

now, rewrite the given polynomial by substituting {{{-x}}} for {{{x}}}:

{{{g(x)=(-x)^4+3(-x)^3+7(-x)^2-6(-x)-13 }}}
{{{g(x)=x^4 -3x^3+7x^2+6x-13 }}}

there are {{{2}}} sign changes for {{{( -x)}}}, which means  there are a {{{maximum}}} of {{{two}}} negative  real zeros

since we have {{{4th}}} degree function,means there are {{{4}}} zeros in all, and we know that complex zeros {{{always}}} come in pairs, it means there will be {{{1}}} negative zero and {{{2}}} imaginary or complex zeros

so, in all will be:

 {{{1}}}  positive real zero

 {{{1}}}  negative real zero

 {{{2}}}  complex zeros