#### Answer

Solution set: $(-\infty, -4)\cup(-1,\infty)$

#### Work Step by Step

Follow the "Procedure for Solving Polynomial lnequalities",\ p.412:
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.
$f(x)= x^{2}+5x+4>0$
factor the trinomial...
find factors of $4$ that add to $5:$
$f(x)=(x+1)(x+4)>0$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$(x+1)(x+4)=0$
$x=-1$ or $x=-4$
3. Locate these boundary points on a number line, thereby dividing the number line into intervals.
4. Test each interval's sign of $f(x)$ with a test value,
$\begin{array}{llll}
Intervals: & (-\infty, -4) & (-4,-1) & (-1,\infty)\\
a=test.val. & -10 & -2 & 0\\
f(a) & (-9)(-6) & (-1)(2) & (1)(4)\\
f(a) > 0 ? & T & F & T
\end{array}$
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: $(-\infty, -4)\cup(-1,\infty)$