<s>[INST] Show that a determinant equals the product of another determinant and a polynomial function without calculating Show without calculating the determinant, that 
$$
    \det\left(\begin{bmatrix}
    a_{1}+b_{1}x & a_{1}-b_{1}x & c_{1}\\
    a_{2}+b_{2}x & a_{2}-b_{2}x & c_{2}\\
    a_{3}+b_{3}x & a_{3}-b_{3}x& c_{3}\\
    \end{bmatrix}\right) 
    = 
    -2x 
    \det\left(\begin{bmatrix}
    a_{1} & b_{1}x & c_{1}\\
    a_{2} & b_{2}x & c_{2}\\
    a_{3} & b_{3}x& c_{3}\\
    \end{bmatrix}\right)
$$ 
I've tried using the multilinearity of the determinant, but i didn't get very far and now im stuck. 
 [/INST] It is indeed multilinearity you want to use. Write
$$
a =
\begin{bmatrix}
a_{1}\\
a_{2}\\
a_{3}
\end{bmatrix},
\quad
b =
\begin{bmatrix}
b_{1}\\
b_{2}\\
b_{3}
\end{bmatrix},
\quad
c =
\begin{bmatrix}
c_{1}\\
c_{2}\\
c_{3}
\end{bmatrix},
$$
then
\begin{align*}
\det(\begin{bmatrix}a + b x & a - b x & c\end{bmatrix})
&=
\det(\begin{bmatrix}a & a & c\end{bmatrix})
- x^{2}
\det(\begin{bmatrix}b & b & c\end{bmatrix})
\\&- x
\det(\begin{bmatrix}a & b & c\end{bmatrix})
+ x
\det(\begin{bmatrix}b & a & c\end{bmatrix}).
\end{align*}
</s>