The mean proportional of (3 + √2) and (12 - √32) is:

√( (3+√2) * (12-4√2) ) = √( (3+√2) * 4(3-√2) ) = √( 4 * (3² - (√2)²) ) = √( 4 * (9-2) ) = √28 = 2√7।