If aligned &. array ccc a^2 & b^2 & c^2 (a+λ)^2 & (b+λ)^2 & (c+λ^2)...

Let \Delta=\left|$\begin{array}{ccc}a^2 & b^2 & c^2 \\ (a+\lambda)^2 & (b+\lambda)^2 & (c+\lambda)^2 \\ (a-\lambda)^2 & (b-\lambda)^2 & (c-\lambda)^2\end{array}$\right| \begin{aligned} &\text { Apply } \mathrm{R}_2 \rightarrow \mathrm{R}_2-\mathrm{R}_3 \\ &\Delta=\left|\begin{array}{ccc} a^2 & b^2 & c^2 \\ (a+\lambda)^2-(a-\lambda)^2 & (b+\lambda)^2-(b-\lambda)^2 & (c+\lambda)^2-(c-\lambda)^2 \\ (a-\lambda)^2 & (b-\lambda)^2 & (c-\lambda)^2 \end{array}\right| \\ &=\left|\begin{array}{ccc} a^2 & b^2 & c^2 \\ 4 a \lambda & 4 b \lambda & 4 c \lambda \\ (a-\lambda)^2 & (b-\lambda)^2 & (c-\lambda)^2 \end{array}\right| \\ &\left(\because(x+y)^2-(x-y)^2=4 x y\right) \\ & \end{aligned}$ Taking out 4 common from $\mathrm{R}_2$ $=4\left|\begin{array}{ccc}a^2 & b^2 & c^2 \\ a \lambda & b \lambda & c \lambda \\ a^2+\lambda^2-2 a \lambda & b^2+\lambda^2-2 b \lambda & c^2+\lambda^2-2 c \lambda\end{array}\right|$ $$\begin{aligned} &\text { Apply } \mathrm{R}_3 \rightarrow\left[\mathrm{R}_3-\left(\mathrm{R}_1-2 \mathrm{R}_2\right)\right] \\ &=4\left|\begin{array}{lll} a^2 & b^2 & c^2 \\ a \lambda & b \lambda & c \lambda \\ \lambda^2 & \lambda^2 & \lambda^2 \end{array}\right| \end{aligned}$$$Taking out$\lambda$common from$\mathrm{R}_2$and$\lambda^2$from$\mathrm{R}_3$.$=4 \lambda\left(\lambda^2\right)\left|$\begin{array}{ccc} a^2 & b^2 & c^2 \\ a & b & c \\ 1 & 1 & 1 \end{array}$\right|$\begin{aligned} &=k \lambda\left|\begin{array}{ccc} a^2 & b^2 & c^2 \\ a & b & c \\ 1 & 1 & 1 \end{array}\right| \\ &\Rightarrow k=4 \lambda^2 \end{aligned}$$$