Sometimes these words mean the same thing, and can be used interchangeably e.g. 'The land extends as far as the river, several kilometres to the north-west' or 'The land stretches as far as the river, several kilometres to the north-west' 'Extended' is more formal, so is more appropriate for a presentation. 'The idea of this proposal could be extended/[stretched] to include the situation in which . . ' 'Stretched' tends to be more of a physical description, as when you stretch an elastic object to make it longer. We can talk about 'stretching' the meaning or purpose of something e.g. 'We can stretch the budget [financial allocation] to include this proposal'. In this case, 'extend' would not be appropriate, as it would imply that the budget could be increased. The expression 'at a stretch' means that something 'can just be managed' : 'We can do it at a stretch, if we have no other problems'.