Q:

Tensor product, vector product

I need help, I don't understand how to check that, the product of the product space given in the form of a distribution in the product base is a product vector. And if it is, calculate tensor product of vectors. I'll be vert grateful for solution with explanation.
$−6e_{1}⊗f_{1}⊗g_{1}+18e_{1}⊗f_{1}⊗g_{2}−10e_{1}⊗f_{2}⊗g_{1}+30e_{1} ⊗f_{2} ⊗g_{2}+ +3e_{2}⊗f_{1} ⊗g_{1}−9e_{2}⊗f_{1}⊗g_{2}+5e_{2}⊗f_{2}⊗g_{1}−15e_{2}⊗f_{2}⊗g_{2}$

A:

$\begin{pmatrix}
a_1\\a_2
\end{pmatrix}
\otimes 
\begin{pmatrix}
b_1\\b_2
\end{pmatrix}
\otimes
\begin{pmatrix}
c_1\\c_2
\end{pmatrix}
$
From (1) $a_1 b_1 c_1 $ contain $2,3$
From (1) (2) $c_2=-3c_1$ 
From (3) $a_1b_2c_1$ contain $2,5$ Thus $b_2=\pm5$, and $b_1=\pm3$ of the same sign, ignore the sign.
$\begin{pmatrix}
a_1\\a_2
\end{pmatrix}
\otimes 
\begin{pmatrix}
3\\5
\end{pmatrix}
\otimes
\begin{pmatrix}
c_1\\c_2
\end{pmatrix}
$
From (4) and (8) 
$a_1=-2a_2$, thus $a_1=-2$, $a_2=1$, $c_1=1$.
$\begin{pmatrix}
-2\\1
\end{pmatrix}
\otimes 
\begin{pmatrix}
3\\5
\end{pmatrix}
\otimes
\begin{pmatrix}
1\\c_2
\end{pmatrix}
$
The rest determined 
$\begin{pmatrix}
-2\\1
\end{pmatrix}
\otimes 
\begin{pmatrix}
3\\5
\end{pmatrix}
\otimes
\begin{pmatrix}
1\\-3
\end{pmatrix}
$
Something like that.

