সরল কর।

প্রদত্ত  রাশি= $\frac{x^2+2x-15}{x^2+x-12}÷\frac{x^2-25}{x^2-x-20}\times \frac{x-2}{x^2-5x+6}$

$=\frac{x^2+5x-3x-15}{x^2+4x-3x-12}÷\frac{{\left(x\right)}^2-{\left(5\right)}^2}{x^2-5x+4x-20}\times \frac{x-2}{x^2-3x-2x+6}$

$=\frac{x(x+5)-3(x+5)}{x(x+4)-3(x+4)}÷\frac{(x+5)(x-5)}{x(x-5)-4(x-5)}\times \frac{(x-2)}{x(x-3)-2(x-3)}$

$=\frac{(x+5)(x-3)}{(x+4)(x-3)}\times \frac{(x+4)(x-5)}{(x+5)(x-5)}\times \frac{(x-2)}{(x-3)(x-2)}$

$=\frac{1}{x-3}$

প্রদত্ত রাশি = $\left(\frac{x}{x-y}-\frac{x}{x+y}\right)÷\left(\frac{y}{x-y}-\frac{y}{x+y}\right)+\left(\frac{x+y}{x-y}+\frac{x-y}{x+y}\right)÷\left(\frac{x+y}{x-y}-\frac{x-y}{x+y}\right)$$=\frac{2xy}{x^2-y^2}÷\frac{2y^2}{x^2-y^2}+\frac{2(x^2+y^2)}{x^2-y^2}÷\frac{4xy}{x^2-y^2}$

$=\frac{2xy}{x^2-y^2}\times \frac{x^2-y^2}{2y^2}+\frac{2(x^2+y^2)}{x^2-y^2}\times \frac{x^2-y^2}{4xy}$

$=\frac{x}{y}+\frac{x^2+y^2}{2xy}$

$=\frac{2x^2+x^2+y^2}{2xy}$

$=\frac{3x^2+y^2}{2xy}$

প্রদত্ত রাশি=$\frac{x^2+2x-3}{x^2+x-2}÷\frac{x^2+x-6}{x^2-4}$

$=\frac{x^2+3x-x-3}{x^2+2x-x-2}÷\frac{x^2+3x-2x-6}{{\left(x\right)}^2-{\left(2\right)}^2}$

$=\frac{x(x+3)-1(x+3)}{x(x+2)-1(x+2)}÷\frac{x(x+3)-2(x+3)}{(x+2)(x-2)}$

$=\frac{(x+3)(x-1)}{(x+2)(x-1)}÷\frac{(x+3)(x-2)}{(x+2)(x-2)}$

$=\frac{(x+3)(x-1)}{(x+2)(x-1)}\times \frac{(x+2)(x-2)}{(x+3)(x-2)}$

=1

প্রদত্ত রাশি = $\frac{a^4-b^4}{a^2+b^2-2ab}\times \frac{{\left(a+b\right)}^2-4ab}{a^3-b^3}÷\frac{a+b}{a^2+ab+b^2}$$=\frac{{\left(a^2\right)}^2-{\left(b^2\right)}^2}{a^2-2ab+b^2}\times \frac{{\left(a-b\right)}^2}{\left(a-b\right)(a^2+ab+b^2)}\times \frac{a^2+ab+b^2}{a+b}$$=\frac{(a^2+b^2)(a^2-b^2)}{{(a-b)}^2}\times \frac{{\left(a-b\right)}^2}{\left(a-b\right)(a^2-ab+b^2)}\times \frac{(a^2+ab+b^2)}{a+b}$$=\frac{(a^2+b^2)(a^2-b^2)}{(a+b)\left(a-b\right)}$

$=\frac{(a^2+b^2)(a^2-b^2)}{\left(a^2-b^2\right)}$

$=(a^2+b^2)$