Calculate: -3x^4+x^2+2 < 0

Expression: $-3{x}^{4}+{x}^{2}+2 < 0$

Write ${x}^{2}$ as a difference

$-3{x}^{4}+3{x}^{2}-2{x}^{2}+2 < 0$

Factor out $-3{x}^{2}$ from the expression

$-3{x}^{2} \times \left( {x}^{2}-1 \right)-2{x}^{2}+2 < 0$

Factor out $-2$ from the expression

$-3{x}^{2} \times \left( {x}^{2}-1 \right)-2\left( {x}^{2}-1 \right) < 0$

Factor out $-\left( {x}^{2}-1 \right)$ from the expression

$-\left( {x}^{2}-1 \right) \times \left( 3{x}^{2}+2 \right) < 0$

Change the signs on both sides of the inequality and flip the inequality sign

$\left( {x}^{2}-1 \right) \times \left( 3{x}^{2}+2 \right) > 0$

Separate the inequality into two possible cases

$\begin{array} { l }\left\{\begin{array} { l } {x}^{2}-1 > 0 \\ 3{x}^{2}+2 > 0\end{array} \right.,\\\left\{\begin{array} { l } {x}^{2}-1 < 0 \\ 3{x}^{2}+2 < 0\end{array} \right.\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\left\{\begin{array} { l } x \in \langle-\infty, -1\rangle \cup \langle1, +\infty\rangle \\ 3{x}^{2}+2 > 0\end{array} \right.,\\\left\{\begin{array} { l } {x}^{2}-1 < 0 \\ 3{x}^{2}+2 < 0\end{array} \right.\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\left\{\begin{array} { l } x \in \langle-\infty, -1\rangle \cup \langle1, +\infty\rangle \\ x \in ℝ\end{array} \right.,\\\left\{\begin{array} { l } {x}^{2}-1 < 0 \\ 3{x}^{2}+2 < 0\end{array} \right.\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\left\{\begin{array} { l } x \in \langle-\infty, -1\rangle \cup \langle1, +\infty\rangle \\ x \in ℝ\end{array} \right.,\\\left\{\begin{array} { l } x \in \langle-1, 1\rangle \\ 3{x}^{2}+2 < 0\end{array} \right.\end{array}$

Solve the inequality for $x$

$\begin{array} { l }\left\{\begin{array} { l } x \in \langle-\infty, -1\rangle \cup \langle1, +\infty\rangle \\ x \in ℝ\end{array} \right.,\\\left\{\begin{array} { l } x \in \langle-1, 1\rangle \\ ∅\end{array} \right.\end{array}$

Find the intersection

$\begin{array} { l }x \in \langle-\infty, -1\rangle \cup \langle1, +\infty\rangle,\\\left\{\begin{array} { l } x \in \langle-1, 1\rangle \\ ∅\end{array} \right.\end{array}$

Find the intersection

$\begin{array} { l }x \in \langle-\infty, -1\rangle \cup \langle1, +\infty\rangle,\\∅\end{array}$

Find the union

$x \in \langle-\infty, -1\rangle \cup \langle1, +\infty\rangle$