Example 13 If the equation $\frac{M x+N}{x^{2}+x-2}$ $=\frac{2}{x+a}-\frac{c}{x+b}$, where $\frac{M x+N}{x^{2}+x-2}$ is in its simplest form, and $a>b, a+b=c$, then $N=$
$$
\begin{array}{l}
\frac{M x+N}{x^{2}+x-2}=\frac{(2-c) x+(2 b-a c)}{x^{2}+(a+b) x+a b} . \\
\left\{\begin{array} { l } 
{ a + b = 1 , } \\
{ a b = - 2 , } \\
{ 2 - c = M , } \\
{ 2 b - a c = N , } \\
{ a > b , a + b = c }
\end{array} \quad \Rightarrow \left\{\begin{array}{l}
a=2, \\
b=-1, \\
c=1, \\
M=1, \\
N=-4 .
\end{array}\right.\right.
\end{array}
$$

The original equation is transformed into
$$
\begin{array}{l}
\frac{M x+N}{x^{2}+x-2}=\frac{(2-c) x+(2 b-a c)}{x^{2}+(a+b) x+a b} . \\
\left\{\begin{array} { l } 
{ a + b = 1 , } \\
{ a b = - 2 , } \\
{ 2 - c = M , } \\
{ 2 b - a c = N , } \\
{ a > b , a + b = c }
\end{array} \quad \Rightarrow \left\{\begin{array}{l}
a=2, \\
b=-1, \\
c=1, \\
M=1, \\
N=-4 .
\end{array}\right.\right.
\end{array}
$$