Find the values of $\alpha $ satisfying the equation(determinant) 
Find the values of $\alpha $ satisfying the equation
  $$\begin{vmatrix}
(1+\alpha)^2 & (1+2\alpha)^2 &   (1+3\alpha)^2\\ 
(2+\alpha)^2& (2+2\alpha)^2 &  (2+3\alpha)^2\\ 
 (3+\alpha)^2& (3+2\alpha)^2 &  (3+3\alpha)^2
\end{vmatrix}=-648\alpha $$

I used $$R_3 \rightarrow R_3- R_2 \qquad R_2 \rightarrow R_2- R_1$$
$$\begin{vmatrix}
(1+\alpha)^2 & (1+2\alpha)^2 &   (1+3\alpha)^2\\ 
3+2\alpha& 3+4\alpha &  3+6\alpha\\ 
 5+2\alpha& 5+4\alpha &  5+6\alpha
\end{vmatrix}$$
Then $$R_3 \rightarrow R_3- R_2$$
$$\begin{vmatrix}
(1+\alpha)^2 & (1+2\alpha)^2 &   (1+3\alpha)^2\\ 
3+2\alpha& 3+4\alpha &  3+6\alpha\\ 
 2& 2 &  2
\end{vmatrix}$$
Now applying column will make zero but the question will become too long . This is contest question and so it should not be that long .

Hint write it as a product of two determinants after taking $\alpha,\alpha^2$ common from one of the determinants to get $\alpha=\pm 9$ or to continue your method use $R_1\to R_1-R_2$
