Explanation

(b)
$3\left( \sin{x - \cos x} \right)^{4} + 6\left( \sin{x + \cos x} \right)^{2} + 4(\sin^{6}{x + \cos^{6}x})$
$= 3\left( 1 - 2\sin x\cos x \right)^{2} + 6\left( 1 + 2\sin x\cos x \right)$
$+ 4\left( \sin^{2}{x + \cos^{2}x} \right)(\sin^{4}x + \cos^{4}x - \sin^{2}x\cos^{2}x)$
$= 3\left\lbrack 1 + 4\sin^{2}{xcos^{2}x - 4\sin x\cos x} \right\rbrack$
$+ 6 + 12\sin x\cos x + 4\lbrack\left( \sin^{2}{x + \cos^{2}x} \right)^{2}$
$- 2\sin^{2}x\cos^{2}x - \sin^{2}x\cos^{2}x\rbrack$
$= 3 + 12\sin^{2}x\cos^{2}{x + 6 + 4 - 12}\sin^{2}x\cos^{2}x$
$= 13$