Question 951325
{{{a/x + (bx + c) /(x^2+3)}}}


{{{(a/x)((x^2+3)/(x^2+3))+((bx+c)/(x^2+3))(x/x)}}}


{{{(a(x^2+3)+x(bx+c))/(x(x^2+3))}}}


{{{(ax^2+3a+bx^2+cx)/(x(x^2+3))}}}


Yes, a squared term should occur.


{{{(3a+cx+(a+b)x^2)/(x(x^2+3))}}}



Now comparing the numerator expressions
{{{x-3=3a+cx+(a+b)x^2}}}
{{{x-3=(a+b)x^2+cx+3a}}}
Equations for the coefficients should give a system,
{{{system(a+b=0,1=c,-3=3a)}}}
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The last equation in the system means {{{highlight(a=-1)}}};  therefore using the first in the system, {{{highlight(b=1)}}}.  The other equation is simply {{{highlight(c=1)}}}.



The decomposed fraction should be {{{highlight(-1/x+(x+1)/(x^2+3))}}}.