This sort of thing should go to an instructor or TA rather than a forum, but I'll show one such.
First, it pays to use code that actually corresponds to the exercise. The value for right was missing a needed term. Second, these are funcions of the independent variable x, not y. I realize it's a dummy variable, but since y is used as dependent variable it makes things very confusing to have definitions using that as the dummy variable; now the sort of thing to do unless one is fairly experienced using the Wolfram Language in calculus.
left[x_] := x*y'[x]
right[x_] := 3*y[x] + x^4*Cos[x]
Third is to recognize that the assignment asks to verify a solution. Call it f[x] for now, and we'll plug it into y to see if it is valid.
f[x_] := x^3*Sin[x]
pluggedIn = left[x] - right[x] /. y -> f
(* Out[309]= -x^4 Cos[x] - 3 x^3 Sin[x] + x (x^3 Cos[x] + 3 x^2 Sin[x]) *)
Is this zero?
Simplify[pluggedIn]
(* Out[310]= 0 *)
