Solve the following: (a + b + c)(ab + bc + ca) - abc = ?

Options

A. (a + b)(b + c)(c - a)
B. (a + b)(b - c)(c + a)
C. (a + b)(b + c)(c + a)
D. (a - b)(b - c)(c - a)

Answer

To solve the expression \((a + b + c)(ab + bc + ca) - abc\), we will follow these steps:

### Step 1: Expand the expression
We start by expanding the product \((a + b + c)(ab + bc + ca)\).

\[ (a + b + c)(ab + bc + ca) = a(ab + bc + ca) + b(ab + bc + ca) + c(ab + bc + ca) \]

### Step 2: Distribute each term
Now, we distribute each term:

1. \(a(ab + bc + ca) = a^2b + abc + aca\)
2. \(b(ab + bc + ca) = ab^2 + b^2c + abc\)
3. \(c(ab + bc + ca) = abc + bca + c^2a\)

Combining these, we have:

\[ a^2b + abc + a^2c + ab^2 + b^2c + abc + abc + c^2a \]

### Step 3: Combine like terms
Next, we combine like terms. Notice that \(abc\) appears three times:

\[ a^2b + ab^2 + a^2c + b^2c + c^2a + 3abc \]

### Step 4: Subtract \(abc\)
Now we subtract \(abc\) from the expression:

\[ (a^2b + ab^2 + a^2c + b^2c + c^2a + 3abc) - abc = a^2b + ab^2 + a^2c + b^2c + c^2a + 2abc \]

### Step 5: Factor the expression
We can factor the expression further. Notice that we can group terms:

\[ = ab(a + b) + ac(a + c) + bc(b + c) + 2abc \]

### Step 6: Final expression
The final expression can be simplified to:

\[ = (a + b)(b + c)(c + a) \]

Thus, the final answer is:

\[ \boxed{(a + b)(b + c)(c + a)} \]