How do you simplify #(x^2+12x+20)/(4x^2-9)*(6x^3-9x^2)/(x^3+10x^2)*(2x+3)#?

Question

Julia Adams · Accepted Answer

#=3(x+2)# #(x^2+12x+20)/(4x^2-9).(6x^3-9x^2)/(x^3+10x^2).(2x+3)# #" "# #=(x^2+12x+20+16-16)/(4x^2-9).(3x^2(2x-3))/(x^2(x+10)).(2x+3)# #" "# #=((x^2+12x+20+16)-16)/(4x^2-9).(3x^2(2x-3))/(x^2(x+10)).(2x+3)# #" "# #=((x^2+12x+36)-16)/(4x^2-9).(3x^2(2x-3))/(x^2(x+10)).(2x+3)# #" "# #=((x^2+2(6)x+6^2)-16)/(4x^2-9).(3x^2(2x-3))/(x^2(x+10)).(2x+3)# #" "# #=((x+6)^2-16)/(4x^2-9).(3x^2(2x-3))/(x^2(x+10)).(2x+3)# #" "# #=((x+6)^2-4^2)/((2x)^2-3^2).(3x^2(2x-3))/(x^2(x+10)).(2x+3)# #" "# Here we will apply the difference of two squares property that says: #" "# #color(blue)(a^2-b^2=(a-b)(a+b)# #" "# #=color(blue)((x+6-4)(x+6+4))/color(blue)((2x-3)(2x+3)).(3x^2(2x-3))/(x^2(x+10)).(2x+3)# #" "# #=((x+2)color(green)cancel((x+10)))/(color(red)cancel((2x-3))color(purple)cancel((2x+3))).(3color(brown)cancel(x^2)color(red)cancel((2x-3)))/(color(brown)cancelx^2color(green)cancel((x+10))).color(purple)cancel((2x+3))# #" "# #=3(x+2)#

Genesis Allen · Answer

undefined To simplify the expression (x^2+12x+20)/(4x^2-9)*(6x^3-9x^2)/(x^3+10x^2)*(2x+3), you can follow these steps:

1. Factorize the numerator and denominator of each fraction separately.
2. Cancel out any common factors between the numerators and denominators.
3. Multiply the remaining factors together to obtain the simplified expression.

Let's break down each fraction:

(x^2+12x+20)/(4x^2-9) can be factored as (x+2)(x+10)/(2x-3)(2x+3).

(6x^3-9x^2)/(x^3+10x^2) can be simplified as 3x^2(2x-3)/(x^2(x+10)).

(2x+3) remains the same.

Now, we can combine the fractions:

[(x+2)(x+10)/(2x-3)(2x+3)] * [3x^2(2x-3)/(x^2(x+10))] * (2x+3)

Next, we can cancel out common factors:

[(x+2)/(2x-3)] * [3x^2/(x^2)] * 1

Simplifying further:

[(x+2)/(2x-3)] * 3

Therefore, the simplified expression is 3(x+2)/(2x-3).