<s>[INST] Finding the inequality satisfying the given conditions. (Multi-choice-multi-correct question):

If the roots of the equation $ax^2+bx+c=0$, $a\ne0$ are imaginary and $a+c<b$,($a,b,c$ are real numbers) then: 
  
  
*
  
*$a+4c< 2b$
  
*$a+b+c<0$
  
*$4a+c<2b$
  
*$4a+c< 2b$ if $a<0$ and $4a+c>2b$ if $a>0$
  

My attempt: 
$b^2-4ac<0 \implies a>0 \land c>0 $ or $a<0 \land c<0$ 
From condition 1 established above, (a>0,c>0) and given condition $a+c<b$
$4a+c+2b>0$ but this isn't the right answer. Where have I gone wrong? Any hints? 
 [/INST] Roots are imaginary numbers $\implies b^2- 4ac < 0\implies b^2 < 4ac \le(a+c)^2\implies b^2 - (a+c)^2 < 0\implies (b-a-c)(b+a+c) < 0$ 
Since $b-a-c > 0$, it must be true that $a+b+c < 0$.
Thus $2)$ is true.
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