がのびるー get longer
が伸 - $$$stretch or physically extend - whether it's a piece of silly putty or a forest that extends all the way to the ocean, or days of summer that get longer.
が延 - prolong : almost never used
を のばす - to stretch or extend intentionally
を延ばす - $$$ to lengthen or extend (the life of a sick person)(the deadline) (your stay in a hotel) also, to procrastinate. More used with time.
を伸ばす - more used with things: stretch your back, extend a collapsible umbrella/telescope/ admantium claw.
延長：えんちょう：extension or prolonging, in time, or in space too ???