For which $m$ does the equation $\lvert x^2+5x+4 \rvert +x-m=0$ have more than two real solutions? When dealing with problems containing quadratic equations, I've never come across one that deals with a number of roots that is bigger than 2.
I've tried breaking up the function into two parts.  
$x^2+6x+4-m=0$ for $x^2+5x+4\ge0$
and
$-x^2-4x-4-m=0$   for $x^2+5x+4<0$
Now imagining the graph of the function $\lvert x^2+5x+4\rvert$, I came to the idea that the equation in the problem should have more than 2 roots if at least one of the roots of the absolute part is smaller than 0 (under the x axis, thus 2+ roots).
The problem is, doing those two inequalities ($f(x_1)<0$ and $f(x_2)<0$), I get that m is greater than $-1$ or $-4$. I'm sure I'm on the right track because a few of the possible intervals contain these two numbers but I'm not sure where I made a mistake.
Edit:
The possible solutions are A: $[1,4]$
B: $[-1,0]$
C: $[0,4]$
D: $[-4,-1]$  
A: I think that drawing the graph of $y=x+|x^2+5x+4|$ will help.
This is because the number of real roots of $|x^2+5x+4|+x-m=0$ equals the number of the intersection points between $y=m$ and $y=x+|x^2+5x+4|$. 
Since $y=m$ is a horizontal line, the graph should tell you the answer. As a result, the answer will be B. I hope this helps.
A: Let $f(x) = \vert x^2+5x+4 \vert + x$. We then have
\begin{align}
f(x) & 
=
\begin{cases}
x^2+6x+4 & x \notin (-4,-1)\\
-(x+2)^2 & x \in (-4,-1)
\end{cases}
\end{align}

Hence, $g(x) = f(x) - m$ shifts the graph down by $m$. We see that $f(x)$ has $3$ roots.


*

*Shifting down any bit will reduce the number of roots to $2$. Hence, $m > 0$ results in two roots for $g(x)$.

*Also, note that shifting up by bringing the sharp peak at $x=-1$ above the $X$ axis but the sharp peak at $x=-4$ remaining below the $X$ axis also produces $2$ roots. The value of $f(x)$ at $x=-1$ is -1, while the value at $x=-4$ is $-4$. Hence, for any $m \in (-4,-1)$ we again have two roots.


Hence, to summarize:


*

*$m>0$ produces $2$ roots.

*$m=0$ produces $3$ roots.

*$m \in (-1,0)$ produces $4$ roots.

*$m=-1$ produces $3$ roots.

*$m \in (-4,-1)$ produces $2$ roots.

*$m=-4$ produces $1$ root.

*$m<-4$ produces no root.
