Discrete math number problem

How would I justify the following statement.

Two integers are consecutive if and only if one is more than the other. Any product of four consecutive integers is one less than a perfect square.

I think this is true.

because for example

$2<3<4<5$

$2\ast 4\ast 5\ast 3=120$

Which one less than 121 a perfect square.

So how would I justify it I did

Let n be a integer

$n(n+1)(n+2)(n+3)+1=(m{)}^{2}$

But I am not sure how to proceed.

How would I justify the following statement.

Two integers are consecutive if and only if one is more than the other. Any product of four consecutive integers is one less than a perfect square.

I think this is true.

because for example

$2<3<4<5$

$2\ast 4\ast 5\ast 3=120$

Which one less than 121 a perfect square.

So how would I justify it I did

Let n be a integer

$n(n+1)(n+2)(n+3)+1=(m{)}^{2}$

But I am not sure how to proceed.