<s>[INST] How many integral values of $k$ such that $2x^3+3x^2+6x+k=0$ has exactly three real roots How many integral values of $k$ such that $2x^3+3x^2+6x+k=0$ has exactly three real roots. I am unable to see how I'd start this question. A small hint, or the entire solution, both will be highly appreciated!
 [/INST] Since this is a cubic polynomial.
By using Vieta's formula we have $a+b+c=-\dfrac{3}{2}$ and $ab+bc+ca=3$. Now squaring $a+b+c$ gives $a^2+b^2+c^2+2(ab+bc+ca)=\dfrac{9}{4}$, so $a^2+b^2+c^2=-\dfrac{15}{4}$ which is always impossible if the roots are real. 
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