matrices - How do you solve a matrix multiplication in which one matrix is unknown?

I'm trying to figure out quadratic interpolation, and this is the only part that I'm struggling with. I'm not too familiar with matrices so explanation of any difficult terms would be helpful.

So let's say I have three matrices:
\begin{equation*}
\begin{bmatrix}a_1&a_2\\a_3&a_4\end{bmatrix}
\begin{bmatrix}b_2\\b_2\end{bmatrix}
=

\begin{bmatrix}c_1\\c_2\end{bmatrix}
\end{equation*}

Which we can label as A,B and C respectively.

If I know the values of all elements within A and C, what is the method used for finding the values within matrix B?