Factorize
`x(y^(2)+z^(2))+y(z^(2)+x^(2))+z(x^(2)+y^(2))+2xy`

Answer

To factorize the expression \( x(y^2 + z^2) + y(z^2 + x^2) + z(x^2 + y^2) + 2xy \), we can follow these steps:

### Step 1: Rewrite the expression
Start by rewriting the expression to group similar terms together:
\[ x(y^2 + z^2) + y(z^2 + x^2) + z(x^2 + y^2) + 2xy \]

### Step 2: Expand the terms
Now, let's expand the terms:
\[ = xy^2 + xz^2 + yz^2 + yx^2 + zx^2 + zy^2 + 2xy \]

### Step 3: Rearrange the terms
Rearranging the terms gives:
\[ = xy^2 + yx^2 + xz^2 + yz^2 + zx^2 + zy^2 + 2xy \]

### Step 4: Group the terms
Next, we can group the terms in pairs:
\[ = (xy^2 + yx^2) + (xz^2 + zx^2) + (yz^2 + zy^2) + 2xy \]

### Step 5: Factor out common factors
Now, we can factor out common factors from each group:
\[ = xy(x + y) + xz(z + x) + yz(y + z) + 2xy \]

### Step 6: Combine like terms
Notice that \( 2xy \) can be written as \( xy + xy \):
\[ = xy(x + y + 1) + xz(z + x) + yz(y + z) \]

### Step 7: Factor the entire expression
Now, we can factor the entire expression:
\[ = (x + y)(y + z)(z + x) \]

Thus, the factorized form of the expression is:
\[ (x + y)(y + z)(z + x) \]