<s>[INST] Show that $2x^6+12x^5+30x^4+60x^3+80x^2+30x+45=0$ has no real roots I tried solving the above question but was unable to prove it.
I used Descartes rule of sign, factorisation techniques, and many other things but could not figure out the solution. 
 [/INST] Factor $x^4$ and separate a non-negative part of the expression covering completely the terms $x^6$ and $x^5,$ then factor $x^2,$ ...
$$\begin{aligned}P(x)=&2x^6+12x^5+30x^4+60x^3+80x^2+30x+45\\=&2x^4(x^2+6x+9)+12x^4+60x^3+80x^2+30x+45\\=&2x^4(x^2+6x+9)+3x^2(4x^2+20x+25)+30x^2+30x+45\\=&2x^4(x^2+6x+9)+3x^2(4x^2+20x+25)+15(2x^2+2x+3)\end{aligned}$$ which is strictly positive for any real $x.$
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