Question 1027248
Expression as given or asked here, 
x-1/(x-3)(x+1)


Appears when rendered,
{{{x-1/(x-3)(x+1)}}}


Raise to equivalent in higher terms,
{{{x((x-3)(x+1))/((x-3)(x+1))-1/(x-3)(x+1)}}}


{{{(x(x^2-2x-3)-1)/((x-3)(x+1))}}}


{{{(x^3-2x^2-3x-1)/((x-3)(x+1))=0}}}
Your interest is now to find the REAL roots of the NUMERATOR.  Start with synthetic division to check on the roots,  -1, and 1, probably the only possible real roots but there MIGHT be irrational roots, also real.


<pre>

-1      |     1    -2    -3    -1
        |          -1    3     0
        |________________________________
             1     -3    0     -1          Not a root


1       |     1     -2     -3    -1
        |
        |            1     -1    -4
        |________________________________
             1      -1     -4     -5       Not a root
</pre>

Use any other methods you know.  You could try a graphing tool.


{{{graph(400,400,-5,5,-10,10,x^3-2x^2-3x-1)}}}
This appears to show two real roots but <s>they are irrational</s>.  This seems strange.


Try a closer look:
{{{graph(400,400,-2,4,-3,3,x^3-2x^2-3x-1)}}}
Only ONE Real, irrational root, and it is slightly greater than 3.