#(8x^2)/(x^2-9) * (x^2 +6x +9) / (16x^3)#
1. Factorising #(x^2 +6x +9)#
We can Split the Middle Term of this expression to factorise it.
#(x^2 +6x +9) = x^2 +3x +3x+9= x (x+3) + 3 (x+3)= color(purple)((x+3) *(x+3)#
2. Factorising #(x^2-9)#:
The above expression is of the form #a^2 - b^2 = (a+b)(a-b)#
So,#(x^2-9) = (x^2 -3^2) = color(blue)((x+3)(x-3)#
The expression now becomes:
#((8x^2)/(color(blue)((x+3)(x-3))))* (color(purple)((x+3) *(x+3)) / (16x^3))#
#=((8x^2)/(color(blue)(cancel((x+3))(x-3))))* (color(purple)(cancel((x+3)) *(x+3)) / (16x^3))#
#=((8x^2)/((x-3))* ((x+3)) / (16x^3))#
#=(cancel(8x^2)/((x-3))* ((x+3)) / (cancel(16x^3)))#
#=((x+3)) / ((x-3) *(2x)#
#=((x+3)) / (2x * (x) + 2x * (-3) #
#=((x+3)) / (2x^2 - 6x) #
