\[2\sin x+2\cos x=\] \[\sqrt{2^2+2^2}\left(\dfrac2{\sqrt{2^2+2^2}}\sin x+\dfrac2{\sqrt{2^2+2^2}}\cos x\right)=\]\[2\sqrt{2}\left(\dfrac1{\sqrt2}\sin x+\dfrac1{\sqrt2}\cos x\right)=\]\[2\sqrt{2}\left(\cos\dfrac{\pi}{4}\sin x+\sin\dfrac{\pi}{4}\cos x\right)=\]\[2\sqrt2\sin\left(x+\dfrac{\pi}{4}\right).\]