Learn Russian: Мы вытащили старый ковёр из-под стола. - We pulled the old carpet from under the table.

The compound preposition из-под (meaning "from under") always requires the Genitive case. For a masculine noun like стол (table), you form the Genitive by adding -а, which gives us стола.

What does the вы- prefix do in the verb вытащили?

The base verb тащить means "to pull" or "to drag". The prefix вы- adds the direction "out", so вытащить means "to pull out". This prefix also makes the verb perfective, emphasizing that the action of pulling the carpet out was successfully completed.

старый ковёр is the direct object being pulled. Why doesn't its ending change?

It actually is in the Accusative case, but because ковёр (carpet) is a masculine inanimate noun, its Accusative form looks exactly like its Nominative (dictionary) form. If they had pulled out an old cat (an animate noun), the endings would change to старого кота.

Why is there a hyphen in из-под?

In Russian, compound prepositions formed with из (like из-под for "from under" and из-за for "from behind") are always written with a hyphen. When you say it out loud, it flows together with the following word, and the stress typically falls on the noun.