We will simplify the expression by rationalizing the denominator, or, in other words, removing the radical from the denominator. We will multiply the expression by the appropriate form of #1# to eliminate the radical while keeping the value of the fraction the same:

#(sqrt(5) + 7)/(sqrt(5) + 7) xx 4/(sqrt(5) - 7) =>#

#(4(sqrt(5) + 7))/((sqrt(5) + 7)(sqrt(5) - 7)) =>#

#(4(sqrt(5) + 7))/((sqrt(5))^2 - 7sqrt(5) + 7sqrt(5) - 7^2) =>#

#(4(sqrt(5) + 7))/(5 - 0 - 49) =>#

#(4(sqrt(5) + 7))/(-44) =>#

#-(4(sqrt(5) + 7))/(4 xx 11) =>#

#-(color(red)(cancel(color(black)(4)))(sqrt(5) + 7))/(color(red)(cancel(color(black)(4))) xx 11) =>#

#-(sqrt(5) + 7)/11#