Show that two summations are equivalent 
Show that
  $$
\sum_{i=1}^n \left(x_i - \bar{x}\right) \left(y_i - \bar{y}\right)
= \left(\sum_{i=1}^n x_i y_i\right) - n \bar{x} \bar{y}.
$$

My work is attached:
.
I'm stuck on what I should do next. 
Any guidance in the right direction would be great!

You are correct. Recall that by definition $\overline{x}=\frac{1}{n}\sum_{i=1}^nx_i$ and  $\overline{y}=\frac{1}{n}\sum_{i=1}^ny_i$.
Therefore by linearity,
$$\sum_{i=1}^n\overline{y}x_i=\overline{y}\sum_{i=1}^nx_i=n\overline{x}\overline{y},\quad
\sum_{i=1}^n\overline{x}y_i=\overline{x}\sum_{i=1}^ny_i=n\overline{x}\overline{y},\quad
\sum_{i=1}^n\overline{x}\overline{y}=\overline{x}\overline{y}\sum_{i=1}^n 1=n\overline{x}\overline{y}.$$
Can you take it from here?
