[ Quadratic inequalities and systems of inequalities ] [ Inequalities. Method of intervals $]$

Find all values of $x$ that satisfy the inequality $(2-a) x^{3}+(1-2 a) x^{2}-6 x+5+4 a-a^{2}<0$ for at least one value of $a$ from the interval $[-1,2]$.

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Let's consider $a$ as an unknown and $x$ as a parameter. Then the question of the problem becomes:

for which values of the parameter $x$ does the quadratic polynomial $f_{x}(a)=a^{2}+\left(x^{3}+2 x^{2}-4\right) a-2 x^{3}-x^{2}+6 x-5$ take at least

one positive value on the interval

$[-1,2] ?$

Obviously, this happens if and only if at least one of the numbers $f_{x}(-1)=6 x-3 x^{3}-3 x^{2}, f_{x}(2)=3 x^{2}$ $+6 x-9$ is positive.

$f_{x}(-1)>0 \Leftrightarrow x(x-1)(x+2)0 \Leftrightarrow x^{2}+2 x-31$.

It remains to combine the solution intervals.

## Answer

$x1$.