$$\eqalign{
& a)\,\,\sin x\left( {2\sin x + 1} \right) = \cos x\left( {2\cos x + \sqrt 3 } \right) \cr
& \Leftrightarrow 2{\sin ^2}x + \sin x = 2{\cos ^2}x + \sqrt 3 \cos x \cr
& \Leftrightarrow \sin x - \sqrt 3 \cos x = 2\cos 2x \cr
& \Leftrightarrow {1 \over 2}\sin x - {{\sqrt 3 } \over 2}\cos x = \cos 2x \cr
& \Leftrightarrow \sin \left( {x - {\pi \over 3}} \right) = \sin \left( {{\pi \over 2} - 2x} \right) \cr
& \Leftrightarrow \left[ \matrix{
x - {\pi \over 3} = {\pi \over 2} - 2x + k2\pi \hfill \cr
x - {\pi \over 3} = \pi - {\pi \over 2} + 2x + k2\pi \hfill \cr} \right. \cr
& \Leftrightarrow \left[ \matrix{
3x = {{5\pi } \over 6} + k2\pi \hfill \cr
x = - {{5\pi } \over 6} + k2\pi \hfill \cr} \right. \Leftrightarrow \left[ \matrix{
x = {{5\pi } \over {18}} + {{k2\pi } \over 3} \hfill \cr
x = - {{5\pi } \over 6} + k2\pi \hfill \cr} \right.\,\,\left( {k \in Z} \right) \cr
& 2)\,\,\left( {2m + 1} \right)\sin x + \left( {2m - 1} \right)\cos x = 2{m^2} + {3 \over 2} \cr
& Xet\,\,hieu\,\,{\left( {2m + 1} \right)^2} + {\left( {2m - 1} \right)^2} - {\left( {2{m^2} + {3 \over 2}} \right)^2} \cr
& \Leftrightarrow 4{m^2} + 4m + 1 + 4{m^2} - 4m + 1 - 4{m^4} - 6{m^2} - {9 \over 4} \cr
& \Leftrightarrow - \left( {4{m^4} - 2{m^2} + {1 \over 4}} \right) = - 4{\left( {{m^2} - {1 \over 4}} \right)^2} \le 0 \cr
& \,{\left( {2m + 1} \right)^2} + {\left( {2m - 1} \right)^2} - {\left( {2{m^2} + {3 \over 2}} \right)^2} < 0\,\, \Rightarrow Pt\,\,vo\,\,nghiem \cr
& {m^2} - {1 \over 4} = 0 \Leftrightarrow m = \pm {1 \over 2} \cr
& TH1:\,\,m = {1 \over 2} \cr
& \Rightarrow 2\sin x = 2 \Leftrightarrow \sin x = 1 \Leftrightarrow x = {\pi \over 2} + k2\pi \,\,\left( {k \in Z} \right) \cr
& TH2:\,\,m = - {1 \over 2} \cr
& \Rightarrow - 2\cos x = 2 \Leftrightarrow \cos x = - 1 \Leftrightarrow x = \pi + k2\pi \,\,\left( {k \in Z} \right) \cr} $$
