The quintic
Once upon a time I tried to understand Galois work in order to understand what specific equations of a given degree admit an algebraic solution...
(Its ending was, that I started to think that the ingrain wallpaper of my study room was actually braille :) ... my desperate attempt of humour aside)
I can't solve your specific problem, but what I do know is, that it is often useful to have a grip on the location of the roots. For instance, we can check for what values the roots are complexx or real.
Since we need to explore polynomials of degree greater than four it is time for NSolve.
First of all, the polynomial:
mypoly[x_, lambda_] := 5 x^5 + 4 x^4 + 3 x^3 + 2 x^2 + x + lambda;
Than the solver:
PolySolve[poly_] := x /. NSolve[poly == 0, x]
So, let's do this for x and lambda = 3:
Clear[x]
PolySolve[mypoly[x, 3]] ==> {-1.,-0.392513-0.877308 I,-0.392513+0.877308 I,0.492513 -0.63794 I,0.492513 +0.63794 I}
Intimidating...Let's produce a plot:
ComplexPlot[x_List, range_List, size_] :=
Module[{r},
r = {Re[#], Im[#]} & /@ x;
ListPlot[r, PlotStyle -> PointSize[size],
AspectRatio -> 1, PlotRange -> {range, range},
PlotRegion -> {{0.05, 0.95}, {0.05, 0.95}}]]
With this we can have a look at the root locus. Here I want to reuse a trick by William T. Shaw. Normally root locus plots are given by joining up the dots to give a smoot curve. This discards the velocity information. His trick to avoid this problem is that he is flattening the two-dimensional list into a one-dimensional of many complex numbers.
ComplexPlot[Flatten[Table[
PolySolve[mypoly[x, lambda]], {lambda, 0, 4, 0.2}]], {-2, 2}, 0.015]

Well. This is the point where I referential refer you, as @OleksandR already did, to Mr. Trott and Mr. Adamchik excellent work...(it is actually quite interesting that if one mathematics taught me, is humility...hardcore)
Please let me know if this helped somehow.
Cheers
Stefan