Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfy $ f\left(m^{2}+m n\right)=f(m)^{2}+f(m) f(n) $ Question -

Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfy the equation
$$
f\left(m^{2}+m n\right)=f(m)^{2}+f(m) f(n)
$$
for all natural numbers $m, n$

by putting $m=1$ and $f(1)=k$ we get 
$f(n+1)=k^2 + kf(n)$

then hint says use $3^2 + 3.1 = 2^2 +2.4$ to get polynomial relation for k..
i am not getting how to use this hint ...i think i am missing some very easy tricks to get at this which i have not lernt yet ...

any help will be appreciated

thankyou
Putting $n=1$ in the condition for $f$ gives
$$f(m^2+m)=f(m)^2+kf(m)$$
Now set $m=3$. By the hint, we have
$$f(3^2+3)=f(2^2+2\cdot 4)=f(2)^2+f(2)f(4)$$
which gives us the condition
$$f(3)^2+kf(3)=f(2)^2+f(2)f(4)$$
You should be able to find $f(2)$,$f(3)$ and $f(4)$ in terms of $k$ by using your condition for $f(n+1)$.

Hope this helps.
The function which you are searching for has the following properties.
$$f(x+y) = f(x)+f(y)$$
and
$$f(x*y) = f(x)*f(y)$$
The only function which fits into these constraints is identity function, i.e,
$$f(x)=x$$