Answer
$$(-∞, -4) ∪ (-1,+∞)$$
Work Step by Step
Let $x^2 + 5x + 4 \gt 0$ be a function $f$.
Find the $x$-intercepts by solving $x^2 + 5x + 4=0$
Factor: $x^2 + 5x + 4$ = $(x+1)(x+4)$
$x+1=0$ or $x+4=0$
$x = -1$ or $x = -4$
These x-intercepts serve as boundary points that separate the number line into intervals.
The boundary points of this equation therefore divide the number line into three intervals:
$(-∞, -4)(-4,-1)(-1, +∞)$
Choose one test value within each interval and evaluate $f$ at that number.
$(-∞, -4)$
Test Value = $-5$
$f=(x+1)(x+4)$
$f=(5+1)(5+4)$
$f=45$
Conclusion: $f (x) > 0$ for all $x$ in $(-∞, -4)$.
$(-4,-1)$
Test Value = $-3$
$f=(x+1)(x+4)$
$f=(-3+1)(-3+4)$
$f=-2$
Conclusion: $f (x) < 0$ for all $x$ in $(-4,-1)$.
$(-1,+∞)$
Test Value = $0$
$f= (x+1)(x+4)$
$f=(0+1)(0+4)$
$f=4$
Conclusion: $f (x) > 0$ for all $x$ in $(-1,+∞)$.
Write the solution set, selecting the interval or intervals that satisfy the given inequality. We are interested in solving $f (x) > 0$, where $f(x) = x^2 + 5x + 4$. Based on the solution above, $f(x)>0$ for all $x$ in $(-∞,-4)$ or $(-1, +∞)$.
Thus, the solution set of the given inequality is:
$$(-∞, -4) ∪ (-1,+∞)$$
