Reduce[7==-4/56-56 c,c]

Expand[1+7x+3x^4-(-1/(56x)-99/784)(-56x+4x^2+5x^3-32x^4)]

Reduce[CoefficientList[(-56x+4x^2+5x^3-32x^4),x][[1;;3]]==CoefficientList[(c1/x+c0)(((233 (x^2))/392)+((47 (x^3))/784)-((51 (x^4))/49)),x][[1;;3]],{c0,c1}]

Expand[(-56x+4x^2+5x^3-32x^4)-(((233 (x^2))/392)+((47 (x^3))/784)-((51 (x^4))/49))(c1/x+c0)/.{c0->(881216/54289),c1->-(21952/233)}]

Reduce[CoefficientList[(((233 (x^2))/392)+((47 (x^3))/784)-((51 (x^4))/49)),x][[1;;4]]==CoefficientList[(c1/x+c0)(-((5104967 (x^3))/54289)-((820064 (x^4))/54289)),x][[1;;4]],{c0,c1}]

Expand[(((233 (x^2))/392)+((47 (x^3))/784)-((51 (x^4))/49))-(-((5104967 (x^3))/54289)-((820064 (x^4))/54289))(c1/x+c0)/.{c0->(22509576625/59567287019632),c1->-(12649337/2001147064)}]

Reduce[CoefficientList[(-((5104967 (x^3))/54289)-((820064 (x^4))/54289)),x][[1;;5]]==CoefficientList[(c1/x+c0)(-((550523087689 (x^4))/531850776961)),x][[1;;5]],{c0,c1}]

Expand[(-((5104967 (x^3))/54289)-((820064 (x^4))/54289))-(-((550523087689 (x^4))/531850776961))(c1/x+c0)/.{c0->(436151675557745504/29887347907548121),c1->(2715080665310265287/29887347907548121)}]

FullSimplify[FromContinuedFraction[{8, -1/(56x)-99/784,(881216/54289)-(21952/(233x)), (22509576625/59567287019632)-(12649337/(2001147064x)),(436151675557745504/29887347907548121)+(2715080665310265287/(29887347907548121x))}]]

ExpandNumerator[%]

Plot[((8+4 (x^2)+5 (x^3)-8 (x^4))/(1+7 x+3 (x^4))),{x,0,5}]

Reduce[-1/(56x)-99/784==0,x]

1+7 x+3 (x^4)/.{x->-14/99}

NSolve[1+7 x+3 (x^4)==0,x]

N[-14/99]

Reduce[(881216/54289)-(21952/(233x))==0,x]

N[(1631/281)]

NSolve[8+4 (x^2)+5 (x^3)-8 (x^4)==0,x]

FullSimplify[Convergents[{8, -1/(56x)-99/784,(881216/54289)-(21952/(233x)), (22509576625/59567287019632)-(12649337/(2001147064x)),(436151675557745504/29887347907548121)+(2715080665310265287/(29887347907548121x))}]]

NSolve[1864+(12860-1163 x) x==0,x]

NSolve[104183+x (712545+4 x (16742+310241 x))==0,x]

Plot[{((8 (14+x))/(14+99 x)),((8 (1864+x (-188+1085 x)))/(1864+(12860-1163 x) x)),((833464+x (-133888+x (1889692+70963 x)))/(104183+x (712545+4 x (16742+310241 x)))),((8+(x^2) (4+(5-8 x) x))/(1+7 x+3 (x^4)))},{x,-4,4}]

ExpandNumerator[((8 (1864+x (-188+1085 x)))/(1864+(12860-1163 x) x))]

ExpandNumerator[FullSimplify[(8-188/233x+1085/233x^2)/(1+3215/466 x-1163/1864x^2)]]