Simplify the expression: $3 \cos^2 x + 4 \sin x \cos x - \sin^2 x - 1$
$3 \cos^2 x + 4 \sin x \cos x - \sin^2 x - 1$
$= 2 \cos^2 x + 2 \times 2 \sin x \cos x - \sin^2 x - \left(1 - \cos^2 x\right)$
$= 2 \cos^2 x + 2 \sin 2 x - \sin^2 x - \sin^2 x$
$= 2 \cos^2 x + 2 \sin 2 x - 2 \sin^2 x$
$= 2 \left(\cos^2 x - \sin^2 x\right) + 2 \sin 2 x$
$= 2 \left(\cos 2 x + \sin 2 x\right)$
$= 2 \sqrt{2} \left(\cos 2 x \times \dfrac{1}{\sqrt{2}} + \sin 2 x \times \dfrac{1}{\sqrt{2}}\right)$
$= 2 \sqrt{2} \left(\sin 2 x \times \cos \dfrac{\pi}{4} + \cos 2 x \times \sin \dfrac{\pi}{4}\right)$
$= 2 \sqrt{2} \sin \left(2x + \dfrac{\pi}{4}\right)$