The number of real roots of the equation

$x^{2}+5|x|+4=0$

for $|x|$, we have 2 cases

case (i)

if $x>0$

Quadratic equation is $x^{2}+5 x+4=0$

i. e. $x^{2}+4 x+x+4=0$

i. e. $(x+4)(x+1)=0$

i.e. $x=-1$ or $-4$

which is not possible since $x>0$

case (ii) for $x<0$,

Quadratic equation is $x^{2}-5 x+4=0$

i. e. $x^{2}-4 x-x+4=0$

i.e. $(x-4)(x-1)=0$

i.e. $x=1$ or $x=4$

which is not possible, since $x$ is negative in this case

Hence, no real solution exist for $x^{2}+5|x|+4=0$