I need help with proving problems My teacher does not want j

I need help with proving problems. My teacher does not want just giving examples that work. He wants me to PROVE it and I am having trouble with it.

For example, He wants us to prove like [Prove that if f(x) and g(x) are both odd functions then the function h(x)=f(x)g(x) is an even function

Need To Show : h(-x)=h(x)

h(-x)=f(-x)g(-x) [by definition of h(x)]

h(-x)=(-f(x))(-g(x)) [by f,g being odd function]

h(-x)=f(x)g(x)

h(-x)=h(x)

Let f(x) = x^2

g(x) = x^3

Lets check whethet f(x)is even or not

f(-x) = (-x)^2 = x^2 ( by defintion f(x) is even)

Lets check g(x) = x^3

g(-x) = (-x)^3 = -x^3

So,g(-x) = - g(x) .Hence g(x) is odd

Now take f( g(x)) : plug x= g(x) in f(x)

f(g(x)) = [x^3]^2 = x^6

let f(g(x)) = h(x) = x^6

h(-x) = (-x)^6 = x^6

h(-x) = h(x)

It is an even function

Hence Proved