Explanation

(b)
Given$\left| \begin{matrix}
x + 1 & 2x + 1 & 3x + 1 \\
2x & 4x + 3 & 6x + 3 \\
4x + 4 & 6x + 4 & 8x + 4
\end{matrix} \right| = 0$
$\Rightarrow 2\left| \begin{matrix}
0 & x & 2x \\
2x & 4x + 3 & 6x + 3 \\
2x + 2 & 3x + 2 & 4x + 2
\end{matrix} \right| = 0$
[Using $(R_{1} \rightarrow 2R_{1} - R_{3})\rbrack$
$\Rightarrow 2\left| \begin{matrix}
0 & x & 0 \\
2x & 4x + 3 & - 2x - 3 \\
2x + 2 & 3x + 2 & - 2x - 2
\end{matrix} \right| = 0$
[Using $(C_{3} \rightarrow C_{3} - {2C}_{2})\rbrack$
$\Rightarrow - 4x\left\lbrack 2x^{2} + 2x - (2x + 3)(x + 1) \right\rbrack = 0$
$\Rightarrow - 4x\left\lbrack 2x^{2} + 2x - \left( 2x^{2} + 5x + 3 \right) \right\rbrack = 0$
$\Rightarrow$ $4x(3x + 3) = 0$
$\Rightarrow x + 1 = 0\ \ \ \lbrack\because x \neq 0\$given$\rbrack$