If $a \mid m$ and $(a + 1) \mid m$, prove $a(a + 1) | m$.
The other answers put this in a general context, but in this example one can be absolutely explicit. If $a\mid m$ and $(a+1)\mid m$ then there are integers $r$ and $s$ such that $$m=ar=(a+1)s.$$ Then $$a(a+1)(r-s)=(a+1)[ar]-a[(a+1)s]=(a+1)m-am=m.$$ As $r-s$ is an integer, then $a(a+1)\mid m$.
If $\rm\,\ a\mid m,\ a\!+\!1\mid m\ \,$ then it follows that $\rm\ \, \color{#90f}{a(a\!+\!1)\mid m}$
${\bf Proof}\rm\quad\displaystyle \frac{m}{a},\; \frac{m}{a+1}\in\mathbb{Z} \ \,\Rightarrow\,\ \frac{m}{a} - \frac{m}{a\!+\!1} \; = \;\color{#90f}{\frac{m}{a(a\!+\!1)} \in \mathbb Z}.\quad$ QED
${\bf Remark}\rm\ \, \text{More generally, if }\, \color{#c00}{n = bc \:\!-\:\! ad} \;$ is a linear combination of $\rm\, a, b\, $ then
$\rm\text{we have}\quad\,\ \displaystyle \frac{m}{a},\; \frac{m}{b}\in\mathbb{Z} \;\;\Rightarrow\;\; \frac{m}{a}\frac{\color{#c00}{bc}}{b} - \frac{\color{#c00}{ad}}{a}\frac{m}{b} = \frac{m\:\!\color{#c00}n}{a\:\!b} \in \mathbb Z$
By Bezout, $\rm\, \color{#c00}{n = \gcd(a,b)}\, $ is the least positive linear combination, so the above yields
$\rm\qquad\qquad a,b\mid m \;\Rightarrow\; ab\mid m\;gcd(a,b) \;\Rightarrow\; \mathfrak{m}_{a,b}\!\mid m\ \ $ for $\ \ \rm \mathfrak{m}_{a,b} := \dfrac{ab}{\gcd(a,b)}$
i.e. $ $ every common multiple $\rm\, m\,$ of $\,\rm a,b\,$ is a multiple of $\;\rm \mathfrak{m}_{a,b},\,$ so $\rm\, \color{#0a0}{\mathfrak{m}_{a,b}\le m}.\,$ But $\rm\,\mathfrak{m}_{a,b}\,$ is also a common multiple, i.e. $\rm\ a,b\mid \mathfrak{m}_{a,b}\,$ viz. $\displaystyle \,\rm \frac{\mathfrak{m}_{a,b}}{a} = \;\frac{a}{a}\frac{b}{gcd(a,b)}\in\mathbb Z\,$ $\,\Rightarrow\,$ $\rm\, a\mid \frak{m}_{a,b},\,$ and $\,\rm b\mid \mathfrak{m}_{a,b}\,$ by symmetry. Thus $\,\rm \mathfrak{m}_{a,b} = lcm(a,b)\,$ is the $\rm\color{#0a0}{least}$ common multiple of $\rm\,a,b.\,$ In fact we have proved the stronger statement that it is a common multiple that is divisibility-least, i.e. it divides every common multiple. This is the general definition of LCM in an arbitrary domain (ring without zero-divisors), i.e. we have the following universal dual definitions of LCM and GCD, which essentially says that LCM & GCD are $\,\sup\,$ & $\,\inf\,$ in the poset induced by divisibility order $\,a\preceq b\!\iff\! a\mid b$.
Definition of LCM $\ \ $ If $\quad\rm a,b\mid c\,\iff\; d\mid c \ \ \,$ then $\rm\ d\approx lcm(a,b)$
compare: $\, $ Def of $\rm\,\cap\ \ \,$ If $\rm\ \ \ a,b\supset c\iff d\supset c\,\ $ then $\,\ \rm d = a\cap b$
Definition of GCD $\ \ $ If $\quad\rm c\mid a,b \;\iff\; c\mid d \,\ $ then $\,\ \rm d \approx \gcd(a,b)$
compare: $\, $ Def of $\rm\,\cup\ \ \,$ If $\rm\ \ \ c\supset a,b\iff c\supset d\,\ $ then $\,\ \rm d = a\cup b$
Note $\;\rm a,b\mid [a,b] \;$ follows by putting $\;\rm c = [a,b] \;$ in the definition. $ $ Dually $\;\rm (a,b)\mid a,b$.
Above $\rm\,d\approx e\,$ means $\rm\,d,e\,$ are associate, i.e. $\rm\,d\mid e\mid d\,$ (equivalently $\rm\,d = u\!\: e\,$ for $\,\rm u\,$ a unit = invertible). In general domains gcds are defined only up to associates (unit multiples), but we can often normalize to rid such unit factors, e.g. normalizing the gcd to be $\ge 0$ in $\Bbb Z,\,$ and making it monic for polynomials over a field, e.g. see here and here.
Such universal definitions enable slick unified proofs of both arrow directions, e.g.
Theorem $\rm\;\; (a,b) = ab/[a,b] \;\;$ if $\;\rm\ [a,b] \;$ exists.
Proof: $\rm\quad d\mid a,b \iff a,b\mid ab/d \iff [a,b]\mid ab/d \iff\ d\mid ab/[a,b] \quad$ QED
The conciseness of the proof arises by exploiting to the hilt the $\:\!(\!\!\iff\!\!)\:\!$ definition of LCM, GCD. Implicit in the above proof is an innate cofactor duality. Brought to the fore, it clarifies LCM, GCD duality (analogous to DeMorgan's Laws), e.g. see here and here.
By the theorem, GCDs exist if LCMs exist. But common multiples clearly comprise an ideal, being closed under subtraction and multiplication by any ring element. Hence in a PID the generator of an ideal of common multiples is clearly an LCM. In Euclidean domains this can be proved directly by a simple descent, e.g. in $\:\mathbb Z \;$ we have the following high-school level proof of the existence of LCMs (and, hence, of GCDs), after noting the set $\rm M$ of common multiples of $\rm a,b$ is closed under subtraction and contains $\:\rm ab \ne 0\:$:
Lemma $\ $ If $\;\rm M\subset\mathbb Z \;$ is closed under subtraction and $\rm M$ contains a nonzero element $\rm\,k,\,$ then $\rm M \:$ has a positive element and the least such positive element of $\;\rm M$ divides every element.
Proof $\, $ Note $\rm\, k-k = 0\in M\,\Rightarrow\, 0-k = -k\in M, \;$ therefore $\rm M$ contains a positive element. Let $\rm\, m\,$ be the least positive element in $\rm\, M.\,$ Since $\,\rm m\mid n \iff m\mid -n, \;$ if some $\rm\, n\in M\,$ is not divisible by $\,\rm m\,$ then we may assume that $\,\rm n > 0,\,$ and the least such. Then $\rm\,M\,$ contains $\rm\, n-m > 0\,$ also not divisible by $\rm m,\,$ and smaller than $\rm n$, contra leastness of $\,\rm n.\ \ $ QED
It is not surprising that you are finding this difficult, because it goes beyond basic divisibility rules -- it rather requires something which is essentially equivalent to the uniqueness of prime factorization. [Edit: Actually this is comment is incorrect -- as Robin Chapman's answer shows, it is possible to prove this using just divisibility rules. In particular it is true in any integral domain.]
I assume $a$ and $m$ are positive integers. The first observation is that $a$ and $a+1$ are relatively prime: i.e., there is no integer $d > 1$ -- or equivalently, no prime number-- which divides both $a$ and $a+1$, for then $d$ would have to divide $(a+1) - a = 1$, so $d = 1$.
Now the key step: since $a$ divides $m$, we may write $m = aM$ for some positive integer $M$. So $a+1$ divides $aM$ and is relatively prime to $a$. I claim that this implies $a+1$ divides $M$. Assuming this, we have $M = (a+1)N$, say, so altogether
$m = aM = a(a+1)N$, so $a(a+1)$ divides $m$.
The claim is a special case of:
(Generalized) Euclid's Lemma: Let $a,b,c$ be positive integers. Suppose $a$ divides $bc$ and $a$ is relatively prime to $b$. Then $a$ divides $c$.
A formal proof of this requires some work! See for instance
http://en.wikipedia.org/wiki/Euclid's_lemma
In particular, proving this is essentialy as hard as proving the fundamental theorem of arithmetic.