M2 Mathematical Induction

For n = k + 1
LHS = 3/4 - (2k + 3)/2(k + 1)(k + 2) + 1/(k + 1)(k + 3)
= 3/4 - 1/(k + 1)[(2k + 3)/2(k + 2) - 1/(k + 3)]
= 3/4 - 1/(k + 1) [ (2k + 3)(k + 3) - 2( k + 2)]/[2(k + 2)(k + 3)]
= 3/4 - [1/2(k + 1)(k + 2)(k + 3)][2k^2 + 9k + 9 - 2k - 4]
= 3/4 - [1/2(k + 1)(k + 2)(k + 3)](2k^2 + 7k + 5)
= 3/4 - [1/2(k + 1)(k + 2)(k + 3)](2k + 5)(k + 1)
= 3/4 - (2k + 5)/2(k + 2)(k + 3)
= 3/4 - [2(k + 1) + 3]/2[(k + 1) + 1][(k + 1) + 2]
= RHS