The set B={1+t2,t+t2,1+2t+t2} is a basis for P2. Find the coordinate vector of p(t)=1+4t+7t2 relative...

Given basis can be written in matrix form (first row - second power, second row - first power, third row - free coefficient):
$B=\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 2\\ 1& 0& 1\end{array}\right]$
Solve $Bx=p(t)$ using augmented matrix:
$\left[\begin{array}{ccc}1& 1& 1|7\\ 0& 1& 2|4\\ 1& 0& 1|1\end{array}\right]$
, multiply first row with -1 and add to third
$\sim \left[\begin{array}{ccc}1& 1& 1|7\\ 0& 1& 2|4\\ 0& -1& 0|-6\end{array}\right]$
${\displaystyle -x_2=-6\to x_2=6}$
${\displaystyle x_2+2x_3=4}$
${\displaystyle 6+2x_3=4}$
${\displaystyle 2x_3=4-6=-2}$
${\displaystyle x_3=-1}$
${\displaystyle x_1+x_2+x_3=7}$
${\displaystyle x_1+6-1=7}$
${\displaystyle x_1=7-5=2}$
$p_B=\left[\begin{array}{c}2\\ 6\\ -1\end{array}\right]$
Results:
$p_B=\left[\begin{array}{c}2\\ 6\\ -1\end{array}\right]$