jmc

Let $$f(t)=x_1(t+1{)}^2+x_2(t+2{)}^2+\cdots +x_7(t+7{)}^2.$$Then the three given equations say $f(0)=1$, $f(1)=12$, and $f(2)=123$, and we want to find $f(3)$.

Since $f(t)$ is a quadratic, we may let $f(t)=A{t}^2+Bt+C$, where $A,B,C$ are constant. Then we have the equations $$\begin{array}{rl}C& =1,\\ A+B+C& =12,\\ 4A+2B+C& =123.\end{array}$$Substituting $C=1$ into the second and third equations gives $A+B=11$ and $4A+2B=122.$ Then $2A+B=61,$ so $A=(2A+B)-(A+B)=61-11=50.$ Then $B=11-A=-39,$ and so $$f(3)=9A+3B+C=9(50)+3(-39)+1=\boxed{334}.$$