Which central idea is shared by both sonnets? the memory of beauty cannot outlast death. time erases beauty in nature and people. writing about people serves to immortalize them. the changes in nature are like the changes in people.

Assuming you are referring to Spenser's Sonnet 75, and Shakespeare's Sonnet 18, the correct answer is writing about people serves to immortalize them.
Both sonnets talk about love - the narrators are writing about their loved ones in order for them to stay alive through poetry and art, even when they die in real life. As long as their poetry exists, the people they wrote about will exist as well - they will be immortal, just like poetry.