Use the following identities:
[tex]\sin{2x}=2\sin{x}\cos{x}\\ \cos{2x}=\cos^2{x}-\sin^2{x} \\ \cos{(x+y)}=\cos{x}\cos{y}-\sin{x}\sin{y} [/tex]
[tex]\sin{2x}+\cos{3x} \\=2\sin{x}\cos{x}+\cos{(2x+x)} \\=2\sin{x}\cos{x}+\cos{2x}\cos{x}-\sin{2x}\sin{x} \\=2\sin{x}\cos{x}+(\cos^2{x}-\sin^2{x})\cos{x}-2\sin{x}\cos{x}\sin{x} \\=2\sin{x}\cos{x}+\cos^3{x}-\sin^2{x}\cos{x}-2\sin^2{x}\cos{x} \\=2\sin{x}\cos{x}+\cos^3{x}-3\sin^2{x}\cos{x}
\\=\sin{x}\cos{x}(2+\cos^2{x}-3\sin{x})[/tex]
