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)
