We are given the rational expression
#color(red)[(2x^2+10x-48)/(8x+64)]#
Denominator(DR) = #(8x+64)#
We will consider the Numerator(NR) first
We have
#color(green)[(2x^2+10x-48)# #color(blue)(NR)#
Rewrite this quadratic expression as
#color(green)[2(x^2+5x-24)#
We can factorize #color(green)[(x^2+5x-24)#:
#rArr x^2 - 3x + 8x - 24#
#rArr x(x-3)+ 8(x-3)#
Hence, we get the factors as
#(x-3) (x+8)#
Hence our #color(blue)(NR)# will now become
#color(green)(2(x-3) (x+8))# #color(blue)(Res.1)#
Next we will consider DR
Denominator(DR) = #(8x+64)#
We can factorize #(8x+64)# as
#color(green)(8(x+8))# #color(blue)(Res.2)#
We will write the rational expression as #color(red)((NR)/(DR))# using our intermediate results #color(blue)(Res.1)# and #color(blue)(Res.2)#
#color(green)(2(x-3) (x+8))/color(green)(8(x+8))#
Simplify to get
#color(green)(2(x-3) cancel((x+8)))/color(green)(8 cancel ((x+8))#
#color(green) (rArr (2(x-3) )/color(green)8#
#color(green) (rArr (cancel 2(x-3) )/color(green)(cancel (8) (color(red)4))#
#color(blue)((x-3)/4)# Our final answer
I hope you find this solution useful.
