Mathematics - Determinant Question with Solution | TestHub

Apply column operations $C_2\to C_2-C_1$ and $C_3\to C_3-C_1$. $$D(x)=\left|\begin{array}{ccc}{x}^2& 2x+1& 4x+4\\ (x+1{)}^2& 2x+3& 4x+8\\ (x+2{)}^2& 2x+5& 4x+12\end{array}\right|$$ Now apply $C_3\to C_3-2C_2$. $$D(x)=\left|\begin{array}{ccc}{x}^2& 2x+1& 2\\ (x+1{)}^2& 2x+3& 2\\ (x+2{)}^2& 2x+5& 2\end{array}\right|$$ Now apply row operations $R_2\to R_2-R_1$ and $R_3\to R_3-R_1$. $$D(x)=\left|\begin{array}{ccc}{x}^2& 2x+1& 2\\ 2x+1& 2& 0\\ 4x+4& 4& 0\end{array}\right|$$ Expand along the third column: $D(x)=2\left|\begin{array}{cc}2x+1& 2\\ 4x+4& 4\end{array}\right|=2[4(2x+1)-2(4x+4)]=2[8x+4-8x-8]=2(-4)=-8$.