<s>[INST] How to solve exercises with polynomial with 2 parameters having all real roots? $f=2x^{4}+4x^{3}+3x^{2}+bx+c$ has all real roots, find b,c (b,c are from R).
Thanks a lot, I tried with substitution, I don't know, is there something with derivative? please help thanks
 [/INST] $f(x) = 2x^4+4x^3+3x^2+bx+c$
If $f(x)$ has all $4$ roots to be real and complex, then it's discriminate $$\Delta = (-72)b^2+176b^3-108b^4-864c-2304bc+1536b^2c+2304c^2-3072bc^2+2048c^3$$ must be real, $\Delta > 0$
Also if you divide $f(x)$ by 2, and make a translation $x = y-\frac{1}{2}$
$2x^4+4x^3+3x^2+bx+c = 0$
$x^4+2x^3+\frac{3/2}x^2+\frac{b/2}x+\frac{c/2} = 0$
Say $x = y-\frac{1}{2}$
The equation is depressed that suddenly appear to be
$y^4+(\frac{b}{2}-\frac{1}{2})y+\frac{c}{2}-\frac{b}{4}+\frac{3}{16} = 0$
Notice that the $y^2$ automatically vanishes
Defined $D = 512c-256b+192$
For the polynomial to therefore have real distinct root $D < 0$ and $\Delta > 0$ 
Nature of roots
</s>