Find the product of the determinants :
`|{:(1,2,1),(-1,0,0),(1,0,1):}|xx|{:(1,-1,0),(0,1,2),(2,-1,1):}|`

Answer

To find the product of the determinants of the two given matrices, we will first calculate the determinant of each matrix separately and then multiply the results.

### Step 1: Calculate the determinant of the first matrix

The first matrix is:
\[ A = \begin{pmatrix} 1 & 2 & 1 \\ -1 & 0 & 0 \\ 1 & 0 & 1 \end{pmatrix} \]

To find the determinant of matrix \(A\), we can use the formula for the determinant of a 3x3 matrix:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is represented as:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

For our matrix \(A\):
- \(a = 1\), \(b = 2\), \(c = 1\)
- \(d = -1\), \(e = 0\), \(f = 0\)
- \(g = 1\), \(h = 0\), \(i = 1\)

Now substituting into the determinant formula:
\[ \text{det}(A) = 1(0 \cdot 1 - 0 \cdot 0) - 2(-1 \cdot 1 - 0 \cdot 1) + 1(-1 \cdot 0 - 0 \cdot 1) \]
\[ = 1(0) - 2(-1) + 1(0) \]
\[ = 0 + 2 + 0 = 2 \]

### Step 2: Calculate the determinant of the second matrix

The second matrix is:
\[ B = \begin{pmatrix} 1 & -1 & 0 \\ 0 & 1 & 2 \\ 2 & -1 & 1 \end{pmatrix} \]

Using the same determinant formula:
For matrix \(B\):
- \(a = 1\), \(b = -1\), \(c = 0\)
- \(d = 0\), \(e = 1\), \(f = 2\)
- \(g = 2\), \(h = -1\), \(i = 1\)

Now substituting into the determinant formula:
\[ \text{det}(B) = 1(1 \cdot 1 - 2 \cdot -1) - (-1)(0 \cdot 1 - 2 \cdot 2) + 0(0 \cdot -1 - 1 \cdot 2) \]
\[ = 1(1 + 2) + 1(0 - 4) + 0 \]
\[ = 1(3) + 1(-4) + 0 = 3 - 4 = -1 \]

### Step 3: Calculate the product of the determinants

Now we multiply the determinants we found:
\[ \text{det}(A) \times \text{det}(B) = 2 \times (-1) = -2 \]

### Final Answer
The product of the determinants is:
\[ \boxed{-2} \]