Question

Prove the following :
$\left|\begin{array}{cccc}1+x& 2& 3& 4\\ 1& 2+x& 3& 4\\ 1& 2& 3+x& 4\\ 1& 2& 3& 4+x\end{array}\right|$ $=x^3(x+10)$.

Answer 1

Apply $C_1+(C_2+C_3+C_4)$ and take out $x+1+2+3+4$ i.e. $x+10$ common from new
$C_1$;