One can of course apply general algorithms for irreducibility testing and factorization, so I presume you are asking if there is something more efficient or more explicit that can be said in the case of trinomials. Except for special cases I don't believe that is the case.
While it is known that every binomial in $\mathbb{Q}(x)$ must have an irreducible factor that is either binomial or trinomial, no analogous bound is known in the trinomial case. It is at least 8 terms due to the known [1] example
$$f(x)f(-x) = - x^{14} - 27180501562500 x^2 + 1244325625000000$$
for $f(x) = x^7 + 20 x^6 + 200 x^5 + 2450 x^4 + 29000 x^3 + 545000 x^2 + 8101250 x + 35275000$
1 Choudhry and A. Schinzel (1992).
On the number of terms in the irreducible factors of a polynomial over $\mathbb Q$.
Glasgow Mathematical Journal (1992), 34, 11-15.
To dig deeper I suggest starting with the work of Schinzel - who has studied these and related factorization problems intensively for almost half a century, e.g. see
MR1254093 (95d:11146) 11R09 (12E05 12E10)
Schinzel, Andrzej.
On reducible trinomials.
Dissertationes Math. (Rozprawy Mat.) 329 (1993), 83 pp.
Let $K$ be a field. It is well known
that a binomial $x^n+a\in K[x]$ is
reducible iff it has the form
$x^{pk}-b^p$ ($p$ prime) or
$x^{4k}+4b^4$. In this treatise the
reducibility of trinomials
$x^n+ax^m+b$ $(a,b\neq 0)$ is
investigated. It turns out that the
situation is very complicated. A
satisfactory answer is obtained if $K$
is a rational function field. For
algebraic function fields in one
variable and for algebraic number
fields, less complete results are
proved. It is assumed throughout that
the characteristic of $K$ does not
divide $mn(n-m)$.
It is easy to find trinomials with
linear or quadratic factors. Table 1
of this paper provides additional
families of reducible trinomials if
$(n,m)$ belongs to a list of 12 pairs,
the largest being $(15,5)$. Perhaps
the simplest example is
$$x^6+4(v+1)x^2-v^2=(x^3+2x^2+2x-v)(x^3-2x^2+2x+v).$$ Every reducible trinomial
$f(x)=x^n+ax^m+b$ gives rise to
additional examples by considering
$u^nf(x^l/u)$ (with $u\in K^\times$
and $l\geq 1$) or $x^nf(1/x)/b$.
Theorem 1 essentially states that
every reducible trinomial arises in
this manner from the examples
indicated before if $K$ is a rational
function field. (More precisely, it is
assumed that $a^{-n}b^{n-m}$ is not a
constant.) Table 2 lists $7$ families
of reducible trinomials
$x^n+A(v,w)x^m+B(v,w)$ with $(v,w)\in
> E(K)$, where $A$, $B$ are polynomials
over $\mathbb Z$ and $E$ is an
elliptic curve defined by an equation
$z^2=C(w)$, where $C$ is a monic
polynomial over $\mathbb Z$. The
polynomials $A,B$ and the
corresponding factorizations of the
trinomials are too complicated to be
included in this review. (For the
largest pair $(n,m)=(21,7)$ the
corresponding $A$ fills 10 lines in
the paper.) In Theorem 2 it is assumed
that $K$ is a finite extension of a
rational function field $F(t)$ such
that $\overline FK$ has genus $g>0$
and $a^{-n}b^{n-m}\notin
> \overline{F}$. If $g=1$ then there are
no additional examples of reducible
trinomials. If $g>1$ then essentially
new examples with $n<24g$ may exist.
Theorem 3 reduces the case where $K$
is a finite separable extension of
$F(t)$ and $a^{-n}b^{n-m}\in\overline
> F$ to studying reducibility over
$K\cap\overline F$. If $K$ is an
algebraic number field then for fixed
$n$, $m$ a finite number of
essentially new examples of reducible
trinomials $x^n+ax^m+b$ may exist
(Theorem 6). The author conjectures
that for every $K$ there is only a
finite number of these ``sporadic
trinomials''. If the conjecture holds
then there exists a constant $c(K)$
such that every trinomial over $K$ has
an irreducible factor with at most
$c(K)$ nonzero coefficients
(Consequence 2). Table 5 contains all
52 sporadic trinomials over $\mathbb
> Q$ known to the author. Their degrees
lie in the range from $8$ to $52$. The
rest of the paper is devoted to
studying the reducibility of
$ax^n+bx^m+c\in\mathbb Z[x]$. Theorem
9 (refining a result of Nagell)
derives necessary conditions, which in
the case $(m,n)=1$ yield an explicit
bound for $b$ in terms of $a,c,m,n$.
For every positive integer $d$ there
exist only finitely many $n,m,b$ with
$n/(m,n)>d$ and $|b|>2$ such that
$x^n+bx^m\pm 1$ has a factor of degree
$d$; and these can be effectively
computed. Theorem 10 derives necessary
conditions from the existence of a
factor (of $ax^n+bx^m+c)$ of given
degree $d$. These imply that there
exists $n_0(d)$ such that $x^n+bx^m+1$
is irreducible if $n\geq n_0(d)$,
$n\neq 2m$, $|b|>2$. By Theorem 8, for
every $n$ there exist only finitely
many reducible trinomials $x^n+bx^m+1$
with $n\neq 2m$.
The proof of Theorem 10 does not
depend on the other results of the
paper. The same applies to Theorem 9.
All other theorems except for Theorem
3 are based on lower estimates for the
genus of certain function fields.
These estimates show that the
existence of a factor of degree $k$ of
$x^n+ax^m+b\in K[x]$ imposes severe
restrictions on $k,m,n,a,b$ provided
$K$ is a function field. The remaining
cases are treated in a long series of
lemmas applying to every field $K$
whose characteristic does not divide
$mn(n-m)$. In several cases the proofs
require extensive manipulations (with
polynomials in several variables)
which were performed by means of
computer algebra systems. Faltings'
theorem (solving Mordell's conjecture)
is invoked in the proof of Theorem 6
(dealing with number fields). Theorems
7 and 8 (concerning
$ax^n+bx^m+c\in\mathbb Z[x])$ are
proved by using the corresponding
theorems for rational function fields
together with a lemma which may be
viewed as a refinement of Hilbert's
irreducibility theorem. The proof of
this lemma is based on Siegel's
theorem (on integral points of curves
of positive genus) and on a result of
Maillet (1919) dealing with rational
functions over $\mathbb Q$ taking
infinitely many integral values at
rational points.
{Reviewer's remarks: In Theorem 2 the
term $u^{\nu-\mu}$ in the expression
for $B$ has to be replaced by $u^\nu$.
The proof of Lemma 27 employs Lemma
2(c) although this lemma only applies
to separable extensions. In order to
prove Lemma 49 one has to know that
every finite separable extension $L$
of $K(t)$ with $L\subseteq \overline
> K(t)$ is contained in $K'(t)$ for some
separable extension $K'$ of $K$. (One
can in fact prove that $L=K'(t)$ for
suitable $K'$. This need not be true
for inseparable $L$.) The proof of
Theorem 6 is apparently based on the
incorrect assumption that a divisor
$P$ of a function field $L=K(t,y)$ has
degree $1$ or is ramified with respect
to $K(t)$ if $t$ and $y$ are congruent
to elements of $K\bmod P$.}
REVISED (1995)
Reviewed by G. Turnwald
