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 $\mathrm{R}_3$.
$$
=4 \lambda\left(\lambda^2\right)\left|\begin{array}{ccc}
a^2 & b^2 & c^2 \\
a & b & c \\
1 & 1 & 1
\end{array}\right|
$$
$$
\begin{aligned}
&=k \lambda\left|\begin{array}{ccc}
a^2 & b^2 & c^2 \\
a & b & c \\
1 & 1 & 1
\end{array}\right| \\
&\Rightarrow k=4 \lambda^2
\end{aligned}
$$