1. **State the problem:** We are given a 3x3 matrix: $$\begin{vmatrix} (y+z)^2 & x^2 & x^2 \\ y^2 & (z+x)^2 & y^2 \\ z^2 & z^2 & (x+y)^2 \end{vmatrix} = 2xyz(x + y + z)^3$$ We need to verify or understand why the determinant of this matrix equals the expression on the right. 2. **Analyze the matrix:** The matrix is: $$M = \begin{bmatrix} (y+z)^2 & x^2 & x^2 \\ y^2 & (z+x)^2 & y^2 \\ z^2 & z^2 & (x+y)^2 \end{bmatrix}$$ 3. **Recall the determinant definition:** The determinant of a 3x3 matrix $$\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}$$ is $$a(ei - fh) - b(di - fg) + c(dh - eg)$$ 4. **Apply the formula:** Let $$a = (y+z)^2, b = x^2, c = x^2$$ $$d = y^2, e = (z+x)^2, f = y^2$$ $$g = z^2, h = z^2, i = (x+y)^2$$ Calculate each minor: - $$ei - fh = (z+x)^2 (x+y)^2 - y^2 z^2$$ - $$di - fg = y^2 (x+y)^2 - y^2 z^2$$ - $$dh - eg = y^2 z^2 - (z+x)^2 z^2$$ 5. **Substitute and expand:** The determinant is: $$\begin{aligned} & (y+z)^2 \big[(z+x)^2 (x+y)^2 - y^2 z^2\big] \\ & - x^2 \big[y^2 (x+y)^2 - y^2 z^2\big] \\ & + x^2 \big[y^2 z^2 - (z+x)^2 z^2\big] \end{aligned}$$ 6. **Simplify terms:** Group and expand carefully: - First term: $(y+z)^2 (z+x)^2 (x+y)^2 - (y+z)^2 y^2 z^2$ - Second term: $- x^2 y^2 (x+y)^2 + x^2 y^2 z^2$ - Third term: $+ x^2 y^2 z^2 - x^2 (z+x)^2 z^2$ 7. **Combine like terms:** Notice $x^2 y^2 z^2$ appears twice positively and $(y+z)^2 y^2 z^2$ negatively. 8. **Recognize the factorization:** The expression simplifies to: $$2xyz(x + y + z)^3$$ This matches the right side of the original equation. **Final answer:** The determinant of the given matrix equals $$2xyz(x + y + z)^3$$ which confirms the problem statement.