Question 1091053
A.

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

create partial fraction template using denominator

{{{4 /((x+2) (x-1)^2)=a[0]/(x+2) +a[1]/(x-1) +a[2]/(x-1)^2}}}

multiply equation by denominator

{{{4((x+2) (x-1)^2) /((x+2) (x-1)^2)=(a[0]((x+2) (x-1)^2))/(x+2) +(a[1]((x+2) (x-1)^2))/(x-1) +(a[2]((x+2) (x-1)^2))/(x-1)^2}}}

simplify

{{{4cross(((x+2) (x-1)^2)) /cross(((x+2) (x-1)^2))=(a[0](cross((x+2)) (x-1)^2))/cross((x+2)) +(a[1]((x+2) (x-1)^cross(2)))/cross((x-1)) +(a[2]((x+2)cross( (x-1)^2)))/cross((x-1)^2)}}}

{{{4= a[0] (x-1)^2 +a[1]((x+2) (x-1)) +a[2](x+2) }}}

solve the unknown by plugging  the real roots of the denominator: {{{1}}} ,{{{ -2}}}

for the root {{{1}}},{{{ a[2]=4/3}}}

for the root {{{-2}}}, {{{a[0]=4/9}}}


plug in the solutions to the known parameters

{{{4= (4/9) (x-1)^2 +a[1]((x+2) (x-1)) +(4/3)(x+2) }}}

{{{4= 4/9(x-1)^2 +a[1]((x+2) (x-1)) +4/3(x+2) }}}

expand

{{{4=4x^2/9+a[1]x^2 +4x/9+a[1]x+28/9-2a[1]}}}

extract variables from within fractions

{{{4= (a[1]x^2 +4x^2/9)+(a[1]x+4x/9)-(2a[1]-28/9)}}}

{{{4=x^2(a[1]+4/9)+x(a[1]+4/9)-(2a[1]-28/9)}}}


solve {{{-(2a[1]-28/9)=4}}} for {{{a[1]}}}

{{{-(2a[1]-28/9)=4}}}

{{{2a[1]-28/9=-4}}}

{{{2a[1]=-4+28/9}}}

{{{2a[1]=-36/9+28/9}}}

{{{2a[1]=-8/9}}}

{{{a[1]=-8/(2*9)}}}

{{{a[1]=-4/9}}}


plug in solutions to the partial fraction parameters to obtain final result

{{{4 /((x+2) (x-1)^2)=(4/9)/(x+2) +(-4/9)/(x-1)+(4/3)/(x-1)^2}}}

simplify
{{{4 /((x+2) (x-1)^2)=4/9(x+2) -4/9(x-1)+4/3(x-1)^2}}}


B.

{{{(5x-1)/ ((x^2+4) (x+1) )}}}

create partial fraction template using denominator

{{{(5x-1)/ ((x^2+4) (x+1) )=a[0]/ (x+1)+(a[2]x+a[1]) /(x^2+4)}}} 

multiply equation by denominator

{{{((5x-1)((x^2+4) (x+1) ))/( ((x^2+4) (x+1) )((x^2+4) (x+1) ))=(a[0]((x^2+4) (x+1) ))/ (x+1)+((a[2]x+a[1])((x^2+4) (x+1) )) /(x^2+4) }}}

simplify

{{{5x-1=a[0](x^2+4) +(a[2]x+a[1]) (x+1) )}}}


solve the unknown by plugging  the real root of the denominator: {{{-1}}} 

{{{5(-1)-1=a[0]((-1)^2+4) +(a[2](-1)+a[1]) (-1+1) )}}}


{{{-5-1=a[0](1+4) +(-a[2]+a[1]) 0 )}}}


{{{-6=5a[0]}}}


{{{ a[0] =-6/5}}}


plug in the solution to the known parameters

{{{5x-1=(-6/5)(x^2+4) +(a[2]x+a[1]) (x+1) )}}}


expand

{{{5x-1= -6x^2/5-24/5 +a[2]x^2+a[2]x+a[1]x+a[1] }}}


extract variables from within fractions

{{{5x-1= (a[2]x^2-6x^2/5)+(a[2]x+a[1]x)+(a[1] -24/5 )}}}


{{{5x-1=x^2 (a[2]-6/5)+x(a[2]+a[1])+(a[1] -24/5 )}}}


use the real root of the denominator({{{ -1}}} ) to create system of equations

{{{a[2]-6/5=0}}} 

{{{a[2]+a[1]=5}}}

{{{a[1] -24/5 =-1}}}

solve the system

{{{a[2]-6/5=0}}} ->{{{a[2]=6/5}}} 

{{{a[1] -24/5 =-1}}}->{{{a[1] =24/5 -1}}}->{{{a[1] =24/5 -5/5}}}->{{{a[1] =19/5}}} 

{{{a[2]+a[1]=5}}}->{{{19/5+6/5=5}}}->{{{25/5=5}}}->{{{true}}}


{{{(5x-1)/ ((x^2+4) (x+1) )=(-6/5)/ (x+1)+((6/5)x+19/5) /(x^2+4) }}}


{{{(5x-1)/ ((x^2+4) (x+1) )=-6/ 5(x+1)+(6x+19) /5(x^2+4) }}}


{{{(5x-1)/ ((x^2+4) (x+1) )=(6x+19) /5(x^2+4)  -6/ 5(x+1)}}}