Let \mathbf x is a (p\times 1) vector, \mathbf\mu_1 is a (p\times 1) vector, \mathbf\mu_2 is a (p\times 1) vector, and \Sigma is a (p\times p) matrix.

Now I have to show

-\frac{1}{2}(\mathbf x-\mathbf\mu_1)'\Sigma^{-1}(\mathbf x-\mathbf\mu_1)+\frac{1}{2}(\mathbf x-\mathbf\mu_2)'\Sigma^{-1}(\mathbf x-\mathbf\mu_2)
= (\mathbf\mu_1-\mathbf\mu_2)'\Sigma^{-1}\mathbf x-\frac{1}{2}(\mathbf\mu_1-\mathbf\mu_2)'\Sigma^{-1}(\mathbf\mu_1+\mathbf\mu_2)

After few lines I got
-\frac{1}{2}(\mathbf x-\mathbf\mu_1)'\Sigma^{-1}(\mathbf x-\mathbf\mu_1)+\frac{1}{2}(\mathbf x-\mathbf\mu_2)'\Sigma^{-1}(\mathbf x-\mathbf\mu_2)

=\frac{1}{2}\mathbf x'\Sigma^{-1}\mathbf\mu_1 + \frac{1}{2}\mathbf\mu_1' \Sigma^{-1}\mathbf x-\frac{1}{2}\mathbf x'\Sigma^{-1}\mathbf\mu_2 - \frac{1}{2}\mathbf\mu_2' \Sigma^{-1}\mathbf x-\frac{1}{2}\mathbf\mu_1' \Sigma^{-1} \mathbf\mu_1 + \frac{1}{2}\mathbf\mu_2' \Sigma^{-1} \mathbf\mu_2

Then I have stumbled to reach the final line

(\mathbf\mu_1-\mathbf\mu_2)'\Sigma^{-1}\mathbf x-\frac{1}{2}(\mathbf\mu_1-\mathbf\mu_2)'\Sigma^{-1}(\mathbf\mu_1+\mathbf\mu_2).