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surely you must mean
(2p^2+10p+12)/(4p^2-4p-48)
= 2(p^2 + 5p + 6)/( 4(p^2 - p - 12) )
= 2(p+2)(p+3)/(4(p-4)(p+3) )
= (p+2)/(2(p-4)) , p ≠-3,4