Let #u =3x^2" "# then #(du)/(dx) =6x# ..............(1)
Let #v =(4x-12)^2" "# then #(dv)/(dx) =8(4x-12) =32x -96#.....(2)
Product #(v (du)/(dx) + u (dv)/(dx))" "#
#color(blue)("~~~~~~~~~Foot Note showing How to obtain (2) ~~~~~~~~~~~~~~")#
#color(brown)("Let "w=4x-12)#
#color(brown)((dw)/(dx)=4)#
#color(brown)("But "v=w^2 " so " (dv)/(dw) = 2w)#
#color(brown)("But "(dv)/(dx) = (dv)/(dw) times (dw)/(dx)) #
#color(brown)("(dv)/(dx)= 2w times 4 = 8w = 8(4x-12))#
#color(blue)("~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")#
By substitution we have:
#{(4x-12)^2 times 6x} + {3x^2 times (32x-96)}#
#color(blue)("I have left the final calculation and simplification for you to complete")#