prove the following algebraically $\left( \begin{array}{c} 2n \\ 2\ \end{array} \right) = 2 \left( \begin{array}{c} n \\ 2\ \end{array} \right) + n^2$ I came across the following proof in my textbook that was used as a end of chapter review. How can I prove the following algebraically? 
$$\left( \begin{array}{c} 2n \\ 2\ \end{array} \right) = 2 \left( \begin{array}{c} n \\ 2\ \end{array} \right) + n^2$$
using $\left( \begin{array}{c} n \\ 2\ \end{array} \right) = \dfrac{n(n-1)}{2}$? 
$$\left( \begin{array}{c} 2n \\ 2\ \end{array} \right) = \frac{2n(2n-1)}{2} = 2n(n-1) = 2 \frac{n(n-1)}{2} + n^2 = 2\left( \begin{array}{c} n \\ 2\ \end{array} \right) + n^2 $$
Is this proof correct? if not how do I fix it?

Since OP asks for an algebraic proof, I type this proof making use of number of roots of a degree two polynomials for fun.
Set 
$f(n) = \left( \begin{array}{c} 2n \\ 2\ \end{array} \right) - 2 \left( \begin{array}{c} n \\ 2\ \end{array} \right) - n^2
$.
Since $\deg f \le 2$, $f$ has at most two distinct roots.
\begin{align}
f(2) &= \left( \begin{array}{c} 4 \\ 2\ \end{array} \right) - 2 \left( \begin{array}{c} 2 \\ 2\ \end{array} \right) - 4 = 6-2-4=0 \\
f(3) &= \left( \begin{array}{c} 6 \\ 2\ \end{array} \right) - 2 \left( \begin{array}{c} 3 \\ 2\ \end{array} \right) - 9 = 15-6-9=0 \\
f(4) &= \left( \begin{array}{c} 8 \\ 2\ \end{array} \right) - 2 \left( \begin{array}{c} 4 \\ 2\ \end{array} \right) - 16 = 28-12-16=0
\end{align}
Therefore, $f$ is identically zero.  Hence $\left( \begin{array}{c} 2n \\ 2\ \end{array} \right) = 2 \left( \begin{array}{c} n \\ 2\ \end{array} \right) + n^2$.
