## Differential Equations and Linear Algebra (4th Edition)

We know that if $a(p_1)(x))+b(p_2)(x))=p(x)\\a(x-4)+b(x^2-x+3)=2x^2-x+2\\bx^2+(a-b)x+(-4a+3b)=2x^2-x+2$, then $b=2,(a-b)=-1,(4a+3b)=2$. Thus $a-b=-1\\a-2=-1\\a=1$ And this doesn't satisfy the last equation because $4(1)+3(2)=3+6=10\ne2$, thus it is not in the span.