B = x ^2 + y^ 2 + 8z^ 2 − 6yz − 6zx + 2xy

It looks like you have a quadratic expression in three variables \( x \), \( y \), and \( z \):

\[
B = x^2 + y^2 + 8z^2 - 6yz - 6zx + 2xy.
\]

To analyze this expression further, we can consider a few approaches, such as completing the square or rewriting it in a more useful form for problem-solving. The goal is to express \( B \) in a way that reveals its properties (like whether it can be minimized or if it represents a particular surface).

### 1. Representing \( B \) in matrix form

The quadratic expression can be rewritten in the standard form using a symmetric matrix. The general form for a quadratic in multiple variables is:

\[
B = \mathbf{X}^T A \mathbf{X}
\]

where \( \mathbf{X} = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \) and \( A \) is a symmetric matrix.

To find the coefficients of this quadratic, we can analyze each term in \( B \):
- The terms involving \( x^2 \), \( y^2 \), and \( z^2 \) from \( B \) form the diagonal of the matrix.
- The coefficients of the mixed terms (like \( xy, xz, yz \)) affect the off-diagonal entries.

The expression can be written in matrix form as:

\[
B = \begin{bmatrix} x & y & z \end{bmatrix} \begin{bmatrix} 1 & 1 & -3 \\ 1 & 1 & -3 \\ -3 & -3 & 8 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}
\]

Where:
- The matrix entries can be inferred from how the variables interact. For instance, the coefficient of \( xy \) is \( 2 \) (hence, both \( A_{12} \) and \( A_{21} \) are \( 1 \)), the coefficient of \( xz \) is \( -6 \) (so, \( A_{13} \) and \( A_{31} \) are \( -3 \)), and so on.

### 2. Completing the Square

We could also complete the square for the given expression, but this process requires careful algebra to combine and reorganize terms appropriately.

### Conclusion

The expression given is a quadratic form in three variables \( x \), \( y \), and \( z \). Whether we analyze this form algebraically, geometrically, or in terms of optimization, it can provide useful insights based on what you're trying to achieve with the expression \( B \).

If you have a specific question or goal regarding this quadratic form, please provide more information, and I will be glad to assist you further!