Answer
$(-3,1)$
Work Step by Step
1. Express the inequality in the form $f(x)<0, f(x)>0, f(x)\leq 0$, or $f(x)\geq 0,$
where $f$ is a polynomial function.
$x^4+3x^3 \lt x+3$,
$x^4+3x^3-x-3 \lt 0$,
$x^3(x+3)-1(x+3) \lt 0$,
$(x^3-1)(x+3) \lt 0$,
$(x^2+x+1)(x-1)(x+3) \lt 0$
$f(x)=(x^2+x+1)(x-1)(x+3)$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$(x^2+x+1)(x-1)(x+3)=0$
$x=1$ or $x=-3$
3. Make a table or diagram: use the test values to make a table or diagram of the sign of each factor in each interval.
4. Test each interval's sign of $f(x)$ with a test value,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \lt 0 ? \\
& &(a^2+a+1)(a-1)(a+3)& \\
(-\infty, -3) & -4 & (+)(-)(-) & F\\
(-3,1) & 0 & (+)(-)(+) & T\\
(1,\infty) & 4 & (+)(+)(+) & F
\end{array}$
5. Write the solution set, selecting the interval or intervals that satisfy the given inequality. If the inequality involves $\leq$ or $\geq$, include the boundary points.
Solution set: $(-3,1)$
