The correct expression that shows how Joshua could use place value to break up the problem of finding the product of 38 × 12 is:
(30 + 8) × (10 + 2)
This expression correctly breaks down the numbers 38 and 12 into their respective place values: 30 (the tens place of 38) and 8 (the ones place of 38), and 10 (the tens place of 12) and 2 (the ones place of 12).