Lagrange interpolation

Lagrange basis.
- $L_0(X)=\frac{(X-1)(X-2)}{(0-1)(0-2)}=\frac{(X-1)(X-2)}{2}$.
- $L_1(X)=\frac{X(X-2)}{(1-0)(1-2)}=-X(X-2)$.
- $L_2(X)=\frac{X(X-1)}{(2-0)(2-1)}=\frac{X(X-1)}{2}$.

Interpolation.
$$P(X)=1\cdot L_0+3\cdot L_1+7\cdot L_2.$$
Compute:
$$P(X)=\frac{(X-1)(X-2)}{2}-3X(X-2)+\frac{7X(X-1)}{2}.$$
Expand: $\frac{X^2-3X+2}{2}-3X^2+6X+\frac{7X^2-7X}{2}=\frac{X^2-3X+2+7X^2-7X}{2}-3X^2+6X=4X^2-5X+1-3X^2+6X=X^2+X+1$.

Check. $P(0)=1$, $P(1)=3$, $P(2)=7$. Correct.

Conclusion. $P(X)=X^2+X+1$.