To solve for $B^{-1} \times C$, we need to follow these steps:

1. Determine $B$ and $C$:

   From the problem, we are given two matrices: $B = \begin{pmatrix} 312 & -6 \\ 321 & -6 \end{pmatrix}$ and $C = \begin{pmatrix} 4 & -10 \\ 253 & 405 \end{pmatrix}$

2. Find the inverse of matrix $B$, $B^{-1}$: To find the inverse of a 2x2 matrix $B = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the formula is: $B^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

   Substituting values from $B$: $B = \begin{pmatrix} 312 & -6 \\ 321 & -6 \end{pmatrix}$ where $a = 312$, $b = -6$, $c = 321$, and $d = -6$.

   The determinant $\text{det}(B)$ is: $\text{det}(B) = (312 \times -6) - (-6 \times 321) = -1872 + 1926 = 54$

   Now, using the inverse formula: $B^{-1} = \frac{1}{54} \begin{pmatrix} -6 & 6 \\ -321 & 312 \end{pmatrix}$ This simplifies to: $B^{-1} = \begin{pmatrix} -\frac{1}{9} & \frac{1}{9} \\ -\frac{107}{18} & \frac{52}{27} \end{pmatrix}$

3. Multiply $B^{-1}$ by $C$: Now that we have $B^{-1}$, we can multiply it by $C$: $B^{-1} \times C = \begin{pmatrix} -\frac{1}{9} & \frac{1}{9} \\ -\frac{107}{18} & \frac{52}{27} \end{pmatrix} \times \begin{pmatrix} 4 & -10 \\ 253 & 405 \end{pmatrix}$

   Perform the matrix multiplication:

   For the first row, first column: $\left(-\frac{1}{9} \times 4 \right) + \left( \frac{1}{9} \times 253 \right) = -\frac{4}{9} + \frac{253}{9} = \frac{249}{9} = 27.67$

   For the first row, second column: $\left(-\frac{1}{9} \times -10 \right) + \left( \frac{1}{9} \times 405 \right) = \frac{10}{9} + \frac{405}{9} = \frac{415}{9} = 46.11$

   For the second row, first column: $\left(-\frac{107}{18} \times 4 \right) + \left( \frac{52}{27} \times 253 \right) = -\frac{428}{18} + \frac{13156}{27} = -23.78 + 487.26 = 463.48$

   For the second row, second column: $\left(-\frac{107}{18} \times -10 \right) + \left( \frac{52}{27} \times 405 \right) = \frac{1070}{18} + \frac{21060}{27} = 59.44 + 780.00 = 839.44$

4. Final Result: The resulting matrix $B^{-1} \times C$ is approximately: $\begin{pmatrix} 27.67 & 46.11 \\ 463.48 & 839.44 \end{pmatrix}$

However, the multiple choices you have given don't exactly match this result. If you meant to provide an approximation, then you might look for values close to the ones we've computed.

Would you like more clarification or assistance on the steps?

Follow-up Questions:

1. Would you like help with matrix multiplication in general?
2. Do you need clarification on how to find the inverse of a matrix?
3. How do you typically approach matrix problems like this?
4. Are you interested in learning about different methods for solving similar problems?
5. Do you need further assistance with matrix operations?

Tip: When calculating the inverse of a matrix, always ensure your determinant is non-zero. If it is zero, the matrix doesn't have an inverse!