Let's simplify the given expression step by step.
We start with:
[tex]\[
\frac{x^2-y^2}{x+y} \times \frac{y^2-3z^2}{y+2} \times m \frac{z^2-x^2}{z+x}
\][/tex]
First, let's factorize the quadratic expressions in the numerators:
[tex]\[
x^2 - y^2 = (x-y)(x+y)
\][/tex]
[tex]\[
y^2 - 3z^2 = (y - \sqrt{3}z)(y + \sqrt{3}z)
\][/tex]
[tex]\[
z^2 - x^2 = (z-x)(z+x)
\][/tex]
Substituting these factorizations back into the expression, we get:
[tex]\[
\frac{(x-y)(x+y)}{x+y} \times \frac{(y - \sqrt{3}z)(y + \sqrt{3}z)}{y+2} \times m \frac{(z-x)(z+x)}{z+x}
\][/tex]
Now we can simplify by cancelling out common factors in the numerator and the denominator:
- In the first term, [tex]\( (x+y) \)[/tex] cancels out:
[tex]\[
(x-y)
\][/tex]
- The second term remains as there are no cancellations possible:
[tex]\[
\frac{(y - \sqrt{3}z)(y + \sqrt{3}z)}{y+2}
\][/tex]
- In the third term, [tex]\( (z+x) \)[/tex] cancels out:
[tex]\[
m(z-x)
\][/tex]
Thus, our simplified expression is:
[tex]\[
(x-y) \times \frac{(y - \sqrt{3}z)(y + \sqrt{3}z)}{y+2} \times m(z-x)
\][/tex]
Combining it all together gives:
[tex]\[
m (x-y)(z-x) \frac{(y - \sqrt{3}z)(y + \sqrt{3}z)}{y+2}
\][/tex]
Finally, rearranging and observing negative sign considerations, the complete and simplified form of the expression is:
[tex]\[
-m \frac{(x^2 - y^2)(x^2 - z^2)(y^2 - 3z^2)}{(x + y)(x + z)(y + 2)}
\][/tex]
This is the fully simplified version of the given mathematical expression.
