2.1.1. (2 points) Find the greatest value of $x$ that satisfies the inequality

$$
\left(6+5 x+x^{2}\right) \sqrt{2 x^{2}-x^{3}-x} \leqslant 0
$$
Answer: 1.

Solution.

$$
\left[\begin{array} { c } 
{ 2 x ^ { 2 } - x ^ { 3 } - x = 0 ; } \\
{ \{ \begin{array} { c } 
{ 2 x ^ { 2 } - x ^ { 3 } - x > 0 ; } \\
{ 6 + 5 x + x ^ { 2 } \leqslant 0 ; }
\end{array} }
\end{array} \Longleftrightarrow \left[\begin{array} { l } 
{ - x ( x - 1 ) ^ { 2 } = 0 ; } \\
{ \{ \begin{array} { c } 
{ - x ( x - 1 ) ^ { 2 } > 0 ; } \\
{ ( x + 2 ) ( x + 3 ) \leqslant 0 ; }
\end{array} }
\end{array} \Longleftrightarrow \left[\begin{array}{l}
x=0 \\
x=1 ; \\
\left\{\begin{array}{l}
x<0 ; \\
-3 \leqslant x \leqslant-2
\end{array}\right.
\end{array}\right.\right.\right.
$$

All solutions of the last system are negative. Therefore, the greatest solution of the original inequality will be 1.