You should actually have
[tex]-x^2-4x-10=-(x^2+4x+10)=-(x^2+4x+4+6)=-((x+2)^2+6)=-(x+2)^2-6[/tex]
Now, remember that [tex]x^2[/tex] is always non-negative, so [tex](x+2)^2\ge0[/tex] for any value of [tex]x[/tex]. This means [tex]-(x+2)^2\le0[/tex] for any [tex]x[/tex], and so
[tex]-(x+2)^2-6\le0-6=-6[/tex]
i.e. f(x) is at most -6, and hence negative for all [tex]x[/tex].
