Let n = the number of sides.

Since it has exactly 1 interior angle that measures 80°, its other n-1
angles must measure 128° each

Those other (n-1) angles must total (n-1)∙128°. 

So the sum of all its interior angles must add up to 80° PLUS (n-1)∙128°,
which is 80°+(n-1)∙128°.

But we also know the formula for the sum of all its interior angles is 
(n-2)∙180°

So we can set them equal:

So we put an equal sign between them:

 80°+(n-1)∙128° = (n-2)∙180° 

Solve that equation for n

Edwin