Calculus of a Single Variable, Chapter 6, 6.1, Section 6.1, Problem 24

To determine whether the given function is a solution of the given differential equation, we first need to find the derivative of the function.
y'=2x(2+e^x)+x^2e^x
Now we plug that into the equation.
x[2x(2+e^x)+x^2e^x]-2x^2(2+e^x)=
2x^2(2+e^x)+x^3e^x-2x^2(2+e^x)=
x^3e^x
As we can see, after simplifying the left hand side we get the right hand side of the equation. This means that the given function is a solution of the given differential equation.
Of course, this is only one of the solutions. The general solution of this equation is y=x^2(c+e^x).
The image below shows graphs of several such functions for different values of c. The graph of the function from the beginning is the green one.