Using properties of determinants, prove that `|[a+x,y,z],[x,a+y,z],[x,y,a+z]|=a^2(a+x+y+z)`

Explanation

To prove that \(|[a+x,y,z],[x,a+y,z],[x,y,a+z]|=a^2(a+x+y+z)\), we will use properties of determinants step by step.

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