Question 1007414
You might really mean this:
(3x-1)/(x+1) in a form a/(x+1)+b


Not the same as what you wrote.



Better notation will resemble the two forms, {{{(3x-1)/(x+1)}}} and {{{a/(x+1)+b}}}.


Begin with the second form, the one using two terms, and combine into a single rational expression.


{{{a/(x+1)+b((x+1)/(x+1))}}}


{{{a/(x+1)+(bx+b)/(x+1)}}}


{{{(a+bx+b)/(x+1)}}}


{{{highlight_green((bx+a+b)/(x+1))}}}, and this is comparable in form to the first given rational expression.  Now the comparison of corresponding expressions or variables and numbers gives you:
{{{system(bx=3x,-1=a+b)}}}


and simpler  {{{system(b=3,a+b=-1)}}}-----b is obvious, and solving for a is simple.  {{{a=-4}}}.