Example 7 When $m=$ $\qquad$, the polynomial
$$
12 x^{2}-10 x y+2 y^{2}+11 x-5 y+m
$$
can be factored into the product of two linear factors.
(1992, Zhengzhou City Junior High School Mathematics Competition)
Solution: First, factorize the quadratic term, we have
$$
12 x^{2}-10 x y+2 y^{2}=(3 x-y)(4 x-2 y) \text {. }
$$

Therefore, we can set
$$
\begin{array}{l} 
12 x^{2}-10 x y+2 y^{2}+11 x-5 y+m \\
=(3 x-y+a)(4 x-2 y+b) \\
=(3 x-y)(4 x-2 y)+(4 a+3 b) x- \\
(2 a+b) y+a b .
\end{array}
$$

Comparing the coefficients of the linear terms, we get
$$
\left\{\begin{array}{l}
4 a+3 b=11, \\
2 a+b=5 .
\end{array}\right.
$$

Solving these, we find $a=2, b=1$.
Comparing the constant term, we get $m=a b=2$.