The two equal signs imply that you have 3 separate 2nd order DEs. It is unlikely that a solution satisfying all 3 does exists.
Assuming just the first and third quantities you can get
In[11]:= DSolve[y''[x] + 10 y'[x] + 4 y[x] == 3 x Cos[x] Sin[x] ,
y[x], x]
Out[11]= {{y[x] ->
E^((-5 - Sqrt[21]) x) C[1] + E^((-5 + Sqrt[21]) x) C[2] - (
3 (-2 Cos[2 x] + 10 x Cos[2 x] - 5 Sin[2 x]))/(
50 (-5 + Sqrt[21])^2 (23 + 5 Sqrt[21]))}}
