Learn French: Le chien est près du coin des enfants. - The dog is near the children's corner.

The phrase for "near" is près de. Since the word for "corner" is masculine (le coin), we need to combine de and le. In French, de + le always contracts into du.

Where does the "apostrophe-S" go in French for "the children's corner"?

French doesn't use an apostrophe-S to show possession. Instead of saying "the children's corner", you have to say "the corner of the children". The word des is a contraction of de (of) and les (the plural). So le coin des enfants literally means "the corner of the children".

Can I ever just say de le or de les instead of memorizing these contractions?

No, the contractions are mandatory in French! Saying de le or de les is grammatically incorrect. However, you don't need to contract feminine words (de la) or words starting with a vowel (de l').

Why might des sound like "dez" when spoken in this sentence?

Usually, the "s" at the end of des is silent. But because enfants starts with a vowel, the "s" links to it and is pronounced like a "z". This is called a liaison, and it makes the sentence flow more smoothly.