(b) Operating ${C_1} \to {C_1} + {C_2} + {C_3}$. We get the value of given determinant as $\left| {\,\begin{array}{*{20}{c}}{3a + 3b}&{a + b}&{a + 2b}\\{3a + 3b}&a&{a + b}\\{3a + 3b}&{a + 2b}&a\end{array}\,} \right|$
= $3\,(a + b)\,\left| {\,\begin{array}{*{20}{c}}1&{a + b}&{a + 2b}\\1&a&{a + b}\\1&{a + 2b}&a\end{array}\,} \right|$
Operate ${R_3} \to {R_3} - {R_1}$, ${R_2} \to {R_2} - {R_1}$
= $3\,(a + b)\,\left| {\,\begin{array}{*{20}{c}}1&{a + b}&{a + 2b}\\0&{ - b}&{ - b}\\0&b&{ - 2b}\end{array}\,} \right|$
$ = 3(a + b)\,(2{b^2} + {b^2}) = 9{b^2}(a + b)$.