Why the leading coefficient is positive? Help is needed in explaining the following (partial) proof:-
Let $Q(x) = ax^4 + bx^3 + cx^2 + dx + e$. 
Suppose “that Q(x) = 0 has no real roots. Thus, Q(x) is always positive or negative for all real x. WLOG, (we can) assume that Q(x) > 0 for all real x, in which case a > 0.”
My question is:- Is the “a > 0” part a further assumption? Or is it a direct consequence of the only assumption. If it is yes to the latter, is there any supporting reason for that?
 A: It follows from the fact that $Q(x)$ is assumed to always be positive. Note that as $x \to \infty$, the term $ax^4$ will become dominant for the value of $Q(x)$.
To be more specific, choose some $x > \max\left\{  \left|4\frac{b}{a}\right| ,\left|4\frac{c}{a}\right| ^{1/2},\left|4\frac{d}{a}\right| ^{1/3},\left|4\frac{e}{a}\right| ^{1/4}\right\}$. Then
$$|ax^4| = |a||x|^4 = \frac{|a|}{4}\left(|x||x|^3+|x|^2|x|^2+|x|^3|x|+|x|^4\right) > \frac{|a|}{4}\left(\left|4\frac{b}{a}\right||x|^3+\left|4\frac{c}{a}\right||x|^2+\left|4\frac{d}{a}\right||x|+\left|4\frac{e}{a}\right|\right) = |bx^3|+|cx^2|+|dx|+|e| > |bx^3+cx^2+dx+e|$$
Hence the sign of $a$ determines the sign of $Q(x)$. Since we assumed $Q(x)>0$, we must have $a>0$.
A: The acronym Without Loss of Generality means that you can hypothetise some property without weakening the generality of the solution, because you can always transform the given problem into one that verifies the property.
In this particular case, if $Q(x)<0$ for all $x$, you can replace $Q(x)$ by $Q'(x)=-Q(x)$ so that $Q'(x)>0$ for all $x$.
A: There are 3 cases to consider: $a>0$, $a<0$ and possibly $a=0$. If the polynomial is of degree 4, the last case is excluded. It then suffices, as already mentioned, to consider $a>0$ (say, of positive type). For non-zero $a$ this is equivalent to ask that $Q>0$ as the leading term dominates when $|x|$ goes to infinity. The map $Q \mapsto -Q$ is a bijection from solutions of positive type to the solutions of negative type.
However, if the polynomial need not be of degree 4, then other solutions are the non-zero constants and quadratic polynomials without roots.
