本题必须展示详细的推理过程。

用代数方法，通过穷举证明：对于所有不是 $3$ 的倍数的正整数 $m$，

的值总是 $3$ 的倍数。（4分）

$m$ 不是 $3$ 的倍数，所以除以 $3$ 的余数只能是 $1$ 或 $2$。因此只需检查 $m=3p+1$ 和 $m=3p+2$ 两类。

Since $m$ is a positive integer that is not a multiple of $3$, there are two cases:

where $p$ is a non-negative integer.

Case 1: $m=3p+1$.

So $m^2+3m+2$ is a multiple of $3$ in this case.

Case 2: $m=3p+2$.

So $m^2+3m+2$ is also a multiple of $3$ in this case.

Therefore, for all positive integers $m$ that are not multiples of $3$, $m^2+3m+2$ is always a multiple of $3$.

先因式分解：$m^2+3m+2=(m+1)(m+2)$。如果 $m$ 不是 $3$ 的倍数，那么接下来的两个整数 $m+1,m+2$ 之中必有一个是 $3$ 的倍数。

Factorise:

Since $m$ is a positive integer not divisible by $3$, $m=3p+1$ or $m=3p+2$, where $p$ is a non-negative integer.

If $m=3p+1$, then

so $m+2$ is a multiple of $3$.

If $m=3p+2$, then

so $m+1$ is a multiple of $3$.

In both cases, one factor of

is a multiple of $3$.

Therefore,

is always a multiple of $3$.