You can try following some recommendations for writing inductive proofs. It's about a different problem, but you can ignore specific details. If you have a difficulty with some step, post here what it is.
LHS = 1*2*3 = 6
RHS = 1(2)(3)(4)/4 = 6 Therefore P(1) is true.
INDUCTIVE STEP: If 1*2*3+...+k(k+1)(k+2)=k(k+1)(k+2)(k+3)/4
then we need to prove
LHS we can rewrite as [k(k+1)(k+2)(k+3)/4] + (k+1)(k+2)(k+3)
= [k(k+1)(k+2)(k+3) + 4(k+1)(k+2)(k+3)]/4 = (k+1)(k+2)(k+3)(k+4)/4
This is equal to the RHS so P(k+1) is true and by MI the original statement is true.
Very good! Only one correction is needed in the equation that needs to be proved in the induction step. The RHS of that equation is the same as the RHS of the induction hypothesis: k(k+1)(k+2)( k+3)/4, while what you obtain in the end is a different (and correct) expression.
If you want to make your instructor completely happy, indicate precisely where you use the induction hypothesis (IH). In your case, this is written as "LHS we can rewrite as...", but it is not clear if this step relies on simple algebra or something else (like the IH). Knowing where the IH is used gives additional confidence that the solution is good. For example, if you never use the IH, was there any reason to use induction at all?