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The image you provided contains a determinant expression that you are asked to prove. The determinant is of a 3x3 matrix, and the equation seems to ask for the expansion and proof of the determinant formula.

The given matrix is:

\[
\text{det} \begin{bmatrix} 
a & b & c \\
a^2 & b^2 & c^2 \\
b+c & c+a & a+b 
\end{bmatrix}
\]

You are asked to prove that the determinant of this matrix is equal to:

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

### Proof Outline:

To prove this, we can perform some row and column operations to simplify the determinant, making it easier to express the result in the form \((a+b+c)(a-b)(b-c)(c-a)\).

### Step 1: Subtract rows or columns
One of the strategies for solving such determinants is to simplify them using elementary row operations without changing the value of the determinant. For instance, you can start by subtracting certain rows or columns, or factorizing terms.

### Step 2: Use known determinant results
This particular form of the determinant suggests it might be related to a Vandermonde determinant or can be simplified using similar algebraic techniques.

### Next Steps
I will now expand the determinant and apply the necessary operations to prove the result. Let me calculate that for you.The determinant of the matrix is:

\[
\text{det} = -a^3b + a^3c + a b^3 - a c^3 - b^3c + b c^3
\]

We can factorize this expression, and upon inspection, it simplifies to:

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

Thus, the determinant of the given matrix indeed equals \((a + b + c)(a - b)(b - c)(c - a)\), proving the desired result.

Would you like a detailed breakdown of the factorization process?

Here are five related questions you could explore further:
1. What is the Vandermonde determinant and how does it relate to this problem?
2. How can you simplify determinants of 3x3 matrices using elementary row and column operations?
3. What are some common properties of determinants that help in solving problems like this?
4. How does the structure of the matrix influence the form of the determinant?
5. Can this method of proof be extended to larger matrices with similar patterns?

**Tip:** When solving determinants, recognizing patterns like the Vandermonde structure can often simplify the calculations significantly.

Prove that the determinant of the given 3x3 matrix is equal to (a+b+c)(a-b)(b-c)(c-a).

This problem proves that the determinant of a 3x3 matrix is equal to (a+b+c)(a-b)(b-c)(c-a). The proof uses matrix algebra and the factorization of determinants, suitable for Grades 11-12 and college students.

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