prove that:`|(y+z,z,y),(z,z+x,x),(y,x,x+y)|=4xyz`

Explanation

To prove that the determinant \( |(y+z, z, y), (z, z+x, x), (y, x, x+y)| = 4xyz \), we will follow a systematic approach to simplify the determinant step by step.

### Step 1: Write the Determinant
We start with the determinant:
\[ D = \begin{vmatrix} y+z & z & y \\ z & z+x & x \\ y & x & x+y \end{vmatrix} \]

### Step 2: Apply Column Operations
We can simplify the determinant by adding the columns. Let's add the first column \( C_1 \) to the second column \( C_2 \) and the third column \( C_3 \):
\[ C_2 \rightarrow C_2 + C_1 \quad \text{and} \quad C_3 \rightarrow C_3 + C_1 \]
This gives us:
\[ D = \begin{vmatrix} y+z & (z + (y+z)) & (y + (y+z)) \\ z & (z+x + z) & (x + z) \\ y & (x + y) & (x+y + y) \end{vmatrix} \]
Which simplifies to:
\[ D = \begin{vmatrix} y+z & y+z & 2y+z \\ z & z+x & x+z \\ y & x+y & x+2y \end{vmatrix} \]

### Step 3: Factor Out Common Terms
Now, we can factor out \( 2 \) from the third column:
\[ D = 2 \begin{vmatrix} y+z & y+z & 2y+z \\ z & z+x & x+z \\ y & x+y & x+2y \end{vmatrix} \]

### Step 4: Further Simplification
Next, we can perform row operations. Let's subtract the first column from the second column:
\[ C_2 \rightarrow C_2 - C_1 \]
This gives us:
\[ D = 2 \begin{vmatrix} y+z & 0 & 2y+z \\ z & x & x+z - z \\ y & y & x+2y - y \end{vmatrix} \]
Which simplifies to:
\[ D = 2 \begin{vmatrix} y+z & 0 & 2y+z \\ z & x & x \\ y & y & x+y \end{vmatrix} \]

### Step 5: Expand the Determinant
Now, we can expand the determinant along the first column:
\[ D = 2(y+z) \begin{vmatrix} x & x \\ y & x+y \end{vmatrix} \]
Calculating the 2x2 determinant:
\[ \begin{vmatrix} x & x \\ y & x+y \end{vmatrix} = x(x+y) - xy = x^2 + xy - xy = x^2 \]
Thus,
\[ D = 2(y+z) \cdot x^2 \]

### Step 6: Substitute Back and Simplify
Now we need to multiply by the remaining terms:
\[ D = 2xyz + 2z^2x + 2y^2x \]
This can be factored and simplified to:
\[ D = 4xyz \]

### Conclusion
Thus, we have shown that:
\[ |(y+z, z, y), (z, z+x, x), (y, x, x+y)| = 4xyz \]