In order to decay to half, it will be 138 days. In order to half that again down to a quarter, it will be another 138 days. 138+138 is 276, thus, 276 days.

276 days

Explanation:

P.S - the exact question is -

To find  - Determine the amount of time for polonium-210 to decay to one fourth its original quantity. The  half-life of polonium-210 is 138 days. Please show all work.

Proof -

Given that , The  half-life of polonium-210 is 138 days

⇒ Amount of polonium -210 to decay to   of its original quantity  = 138 days

Now,

Amount of polonium to decay to next    = 138 days

∴ we get

Total Amount of polonium -210 to decay to   of its original quantity  =138 + 138 days

                                                                                                   = 276 days

⇒Total amount of time for polonium-210 to decay to one fourth its original quantity = 276 days.

pretty sure it's false