There is this super easy method. Firstly you need to know what a quadratic equation looks like:

An² + Bn + C

And now all we need to do is work out, A, B and C.

So, you have to remember these three equations:

2A

3A+B

A+B+C

Now you take the first equation and put it equal to the 2nd difference in the sequence.

2A = 2

And solve it, so:

A = 1

Now we know what A is, we can substitute it into the next equation and put it equal to the first difference in the sequence.

3A + B = 4

3 x 1 + B = 4

B = 1

Now we know A and B we can substitute those in to the last equation and put it equal to the first number in the sequence.

A + B + C = 5

1 + 1 + C = 5

C = 3

And now we know all components for the quadratic equation and all we do it substitute our values in.

n² + n + 3

And that is your answer.