Without using a calculator, express (5+2√3)22+√3(5+2√3)22+√3 in the form p+q√3p+q√3, where pp and qq are integers.
(5+2√3)22+√3(5+2√3)22+√3
=(25+20√3+12)(2−√3)(2+√3)(2−√3)=(25+20√3+12)(2−√3)(2+√3)(2−√3)
=(37+20√3)(2−√3)4−3=(37+20√3)(2−√3)4−3
=74−37√3+40√3−60=74−37√3+40√3−60
=14+3√3=14+3√3