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Please solve the following determinant based question | (y+z)^2 xy zx |

| xy (x+z)^2 yz | = 2xyz(x+y+z)^3 .

| xz yz (x+y)^2 |

Please give the answer fast !!

$| \begin{matrix} (y + z)^{2} & x y & x z \\ x y & (x + z)^{2} & y z \\ x z & y z & (x + y)^{2} \end{matrix} | M u l t i p l y R_{1}, R_{2} a n d R_{3} b y x y z \frac{1}{x y z} | \begin{matrix} x (y + z)^{2} & x^{2} y & x^{2} z \\ x y^{2} & (x + z)^{2} y & y^{2} z \\ x z^{2} & y z^{2} & z (x + y)^{2} \end{matrix} | = \frac{x y z}{x y z} | \begin{matrix} (y + z)^{2} & x^{2} & x^{2} \\ y^{2} & (x + z)^{2} & y^{2} \\ z^{2} & z^{2} & (x + y)^{2} \end{matrix} | A p p l y i n g C_{1} \to C_{1} - C_{3} a n d C_{2} \to C_{2} - C_{3} | \begin{matrix} (y + z)^{2} - x^{2} & 0 & x^{2} \\ 0 & (x + z)^{2} - y^{2} & y^{2} \\ z^{2} - (x + y)^{2} & z^{2} - (x + y)^{2} & (x + y)^{2} \end{matrix} | = | \begin{matrix} (y + z + x) (y + z - x) & 0 & x^{2} \\ 0 & (x + z + y) (x + z - y) & y^{2} \\ (z + x + y) (z - x - y) & (z - x - y) (z + x + y) & (x + y)^{2} \end{matrix} | T a k i n g (x + y + z) c o m m o n f r o m C_{1} a n d C_{2}, (x + y + z)^{2} | \begin{matrix} (y + z - x) & 0 & x^{2} \\ 0 & (x + z - y) & y^{2} \\ (z - x - y) & (z - x - y) & (x + y)^{2} \end{matrix} | A p p l y i n g R_{3} \to R_{3} - (R_{1} + R_{2}) (x + y + z)^{2} | \begin{matrix} (y + z - x) & 0 & x^{2} \\ 0 & (x + z - y) & y^{2} \\ (z - x - y) & (z - x - y) & (x + y)^{2} \end{matrix} | = (x + y + z)^{2} | \begin{matrix} (y + z - x) & 0 & x^{2} \\ 0 & (x + z - y) & y^{2} \\ - 2 y & - 2 x & 2 x y \end{matrix} | A p p l y i n g C_{1} \to x C_{1}, C_{2} \to y C_{2} \frac{(x + y + z)^{2}}{x y} | \begin{matrix} x y + x z - x^{2} & 0 & x^{2} \\ 0 & x y + z y - y^{2} & y^{2} \\ - 2 x y & - 2 x y & 2 x y \end{matrix} | A p p l y i n g C_{1} \to C_{1} + C_{3}, C_{2} \to C_{2} + C_{3} \frac{(x + y + z)^{2}}{x y} | \begin{matrix} x y + x z & x^{2} & x^{2} \\ y^{2} & x y + z y & y^{2} \\ 0 & 0 & 2 x y \end{matrix} | T a k i n g x, y a n d 2 x y c o m m o n f r o m R_{1}, R_{2}, R_{3} r e s p e c t i v e l y . \frac{(x + y + z)^{2}}{x y} \times x \times y \times 2 x y | \begin{matrix} y + z & x & x \\ y & x + z & y \\ 0 & 0 & 1 \end{matrix} | = 2 x y (x + y + z)^{2} | \begin{matrix} y + z & x & x \\ y & x + z & y \\ 0 & 0 & 1 \end{matrix} | E x p a n d i n g a l o n g R_{3}, 2 x y (x + y + z)^{2} [(y + z) (x + z) - x y] = 2 x y (x + y + z)^{2} [z (x + y + z)] = 2 x y z (x + y + z)^{3} = R H S$

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