I couldn't see the picture clearly, so I wrote it down according to what I saw and understood. But the method is still the same.

Solution:

Because f(x) is the odd function on r, for any x∈R, there are:

f(-x)=-f(x)；

f(0)= 0；

By the question, when x∈0, +∞),

F(x)=x( 1+3√x), (where 3 is a corner marker. I can't see clearly. . . )

Satisfy f(0)=0.

When x∑(-∞, 0), -x∑(0, ∞),

So f(-x)=-x[ 1+3√(-x)].

And f(-x)=-f(x), so:

f(x)=-f(-x)=x[ 1+3√(-x)]

This is what you want!