Answer
Should I extend the house or build a garden room?
Garden room for a separate, cheaper, quicker space; extension for connected living space.
In short
A garden room is usually quicker, less disruptive and cheaper than extending, and often permitted development, while an extension adds connected space to the house itself. If you want a separate office, studio or gym, a garden room is often the better answer; for connected living space, an extension.
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