Answer
$$p(t)=7+6t-t^2$$
Work Step by Step
The system can be writen in this augmented form:
$$\left[\begin{array}{rrr}
1 &1 &1^2 &12 \\
1 &2 & 2^2 & 15 \\
1 &3 & 3^2 & 16
\end{array}\right]$$
By calculation, the reduced echelon form is:
$$\left[\begin{array}{rrr}
1 &0 &0 &7 \\
0 &1 & 0 & 6 \\
0 &0 & 1 & -1
\end{array}\right]$$
The solution is $(7,6,-1)$, which corresponds to $(a_0,a_1,a_2)$. Substitute $(a_0,a_1,a_2)$ with $(7,6,-1)$ in $p(t)=a_0+a_1t+a_2t^2$.
