Expand[4+5x+8x^2+x^3-(4/11)(11-6x+4x^2)]

CoefficientList[(4+5x+8x^2+x^3)(4/11)-((16/79+C1*x)/x)(((79 x)/11)+((72 (x^2))/11)+(x^3)),x]

Solve[(428/869)-((79 C1)/11)==0]

Expand[(4+5x+8x^2+x^3)(4/11)-((16/79+C1*x)/x)(((79 x)/11)+((72 (x^2))/11)+(x^3))/.{C1->(428/6241)}]

CoefficientList[x((4+5x+8x^2+x^3)(4/11)((16/79+(428/6241)*x)/x)-((C3+C2 x+ C1 x^2 + C0 x^3)/x^3)*(((154992 (x^2))/68651)+((20256 (x^3))/68651))),x]

ToRules[Reduce[{(256/869)-((154992 C3)/68651),(32128/68651)-((154992 C2)/68651)-((20256 C3)/68651),(49008/68651)-((154992 C1)/68651)-((20256 C2)/68651),(18752/68651)-((154992 C0)/68651)-((20256 C1)/68651)}=={0,0,0,0},{C3,C2,C1,C0}]]

Expand[((4+5x+8x^2+x^3)(4/11)((16/79+(428/6241)*x)/x)-((C3+C2 x+ C1 x^2 + C0 x^3)/x^3)*(((154992 (x^2))/68651)+((20256 (x^3))/68651)))/.{C3->(1264/9687),C2->(5950424/31279323),C1->(29425109855/101000933967),C0->(27040301844298/326132015779443)}]

q1=4/11
q2=((16/79+C1*x)/x)/.{C1->(428/6241)}
q3=((C3+C2 x+ C1 x^2 + C0 x^3)/x^3)/.{C3->(1264/9687),C2->(5950424/31279323),C1->(29425109855/101000933967),C0->(27040301844298/326132015779443)}

CoefficientList[x^4*((4+5x+8x^2+x^3)*q1*q2*q3-((c7+c6 x+ c5 x^2+ c4 x^3+c3 x^4+c2 x^5+c1 x^6+c0 x^7)/x^7)*((3536552285435376 (x^3))/7463096338424847131)),x]

ToRules[Reduce[{(4096/106557)-((3536552285435376 c7)/7463096338424847131),(1061056512/9060577229)-((3536552285435376 c6)/7463096338424847131),(1858275696371200/6933815117768517)-((3536552285435376 c5)/7463096338424847131),(7437996882306872576/22389289015274541393)-((3536552285435376 c4)/7463096338424847131),(6761442294242207824/22389289015274541393)-((3536552285435376 c3)/7463096338424847131),(3213103101959260832/22389289015274541393)-((3536552285435376 c2)/7463096338424847131),(223241053289329712/7463096338424847131)-((3536552285435376 c1)/7463096338424847131),(46292996757438176/22389289015274541393)-((3536552285435376 c0)/7463096338424847131)}=={0,0,0,0,0,0,0,0},{c0,c1,c2,c3,c4,c5,c6,c7}]]

q4=((c7+c6 x+ c5 x^2+ c4 x^3+c3 x^4+c2 x^5+c1 x^6+c0 x^7)/x^7)/.{c0->(2893312297339886/663103553519133),c1->(13952565830583107/221034517839711),c2->(200818943872453802/663103553519133),c3->(422590143390137989/663103553519133),c4->(464874805144179536/663103553519133),c5->(375023263973912800/663103553519133),c6->(6069308378351872/24559390871079),c7->(53789596065113344/663103553519133)}

q1
ExpandNumerator[Together[Expand[q2]]]
ExpandNumerator[Together[Expand[q3]]]
ExpandNumerator[Together[Expand[q4]]]

ExpandNumerator[Together[Expand[1/q1-1/q2+1/q3-1/q4]]]

Plot[{((11-6 x+4 (x^2))/(4+5 x+8 (x^2)+(x^3))), 1/q1-1/q2+1/q3-1/q4},{x,-10,10}]

Together[Expand[1/q1-1/q2+1/q3-1/q4]]