for what values of x:
`[1" "2" "1][{:(1,2,0),(2,0,1),(1,0,2):}][{:(0),(2),(x):}]=0? `

Explanation

To solve the equation given by the matrix multiplication, we need to find the values of \( x \) such that:

\[ [1 \quad 2 \quad 1] \begin{pmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{pmatrix} \begin{pmatrix} 0 \\ 2 \\ x \end{pmatrix} = 0 \]

### Step 1: Perform the matrix multiplication

First, we multiply the second matrix by the third matrix:

\[ \begin{pmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{pmatrix} \begin{pmatrix} 0 \\ 2 \\ x \end{pmatrix} \]

Calculating each row:

1. For the first row:
\[ 1 \cdot 0 + 2 \cdot 2 + 0 \cdot x = 0 + 4 + 0 = 4 \]

2. For the second row:
\[ 2 \cdot 0 + 0 \cdot 2 + 1 \cdot x = 0 + 0 + x = x \]

3. For the third row:
\[ 1 \cdot 0 + 0 \cdot 2 + 2 \cdot x = 0 + 0 + 2x = 2x \]

Thus, the result of the multiplication is:

\[ \begin{pmatrix} 4 \\ x \\ 2x \end{pmatrix} \]

### Step 2: Multiply the result by the first matrix

Now we multiply the first matrix by the result:

\[ [1 \quad 2 \quad 1] \begin{pmatrix} 4 \\ x \\ 2x \end{pmatrix} \]

Calculating this gives:

\[ 1 \cdot 4 + 2 \cdot x + 1 \cdot 2x = 4 + 2x + 2x = 4 + 4x \]

### Step 3: Set the equation to zero

Now we set the result equal to zero:

\[ 4 + 4x = 0 \]

### Step 4: Solve for \( x \)

To find \( x \), we can rearrange the equation:

\[ 4x = -4 \]

Dividing both sides by 4 gives:

\[ x = -1 \]

### Conclusion

The value of \( x \) that satisfies the equation is:

\[ \boxed{-1} \]