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How do you divide #\frac { 6p ^ { 2} + p - 12} { 8p ^ { 2} + 18p + 9} \div \frac { 6p ^ { 2} - 11p + 4} { 2p ^ { 2} + 13p - 7}#?

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#(p-7)/(4p+3)#

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Let's begin by factorizing each quadratic trinomial.

#=>6p^2+p−12#
#=6p^2+9p-8p−12#
#=3p(2p+3)-4(2p+3)#
#=color(red)((3p-4)(2p+3))#

#=>8p^2+18p+9#
#=8p^2+6p+12p+9#
#=2p(4p+3) + 3(4p+3)#
#=color(blue)((4p+3)(2p+3))#

#=>6p^2−11p+4#
#=6p^2−3p-8p+4#
#=3p(2p-1)-4(2p-1)#
#=color(green)((3p-4)(2p-1))#

#=>2p^2+13p−7#
#=2p^2-1p+14p-7#
#=p(2p-1) -7(2p-1)#
#=color(magenta)((p-7)(2p-1))#

now let's combine everything.
the question is,
#color(red)(6p^2+p−12)/color(blue)(8p^2+18p+9) -: color(green)(6p^2−11p+4)/color(magenta)(2p^2+13p−7)#

#=> color(red)(6p^2+p−12)/color(blue)(8p^2+18p+9) xx color(magenta)(2p^2+13p−7)/color(green)(6p^2−11p+4)#

#=>color(red)(cancel((3p-4))cancel((2p+3)))/color(blue)((4p+3)cancel((2p+3))) xx color(magenta)((p-7)cancel((2p-1)))/color(green)(cancel((3p-4))cancel((2p-1)))#

#=>(p-7)/(4p+3)#

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