Find one & all polynomial of degree 4

If someone can advise me how to approach the following problem it would be greatly appreciated.

Q: Find one & all polynomials of degree 4 such that p(-1)=1, p(0)=0, p(1)=2, p(2)=1

I tried...

p(x)=ax^4+bx^3+cx^2+dx+e

p(-1)=a-b+c-d+e=1

p(0)=e=0

p(1)=a+b+c+d+e=2

p(2)=16a+8b+4c+2d+e=1

Is it correct to place it into a matrix and reduce to echelon form? But then I'm not sure what to do with it.

[1 1 1 1 2] [1 1 1 1 2]

[1 -1 1 -1 1] ~ [0 -2 0 -2 -1]

[16 8 4 2 1] [0 0 1 1/2 -9/4]