Any #P_2(x)=a x^2+bx+c# can be represented as
#alpha p_1+beta p_2+gammap_3+delta p_4#
or
#a x^2+bx+c=alpha+beta -(3beta-gamma)x+delta x^2#
with
#{(alpha+beta=a),(-3beta+gamma=b),(delta=c):}#
or
#((0,0,0,1),(0,-3,1,0),(1,1,0,0))((alpha),(beta),(gamma),(delta))=((a),(b),(c))#
or
#((0,0,1),(0,-3,0),(1,1,0))((alpha),(beta),(delta))=((a),(b),(c))-gamma((0),(1),(0))#
Here #gamma# is a free parameter and can be set to #0#. Also
#((0,0,1),(0,-3,0),(1,1,0))# is invertible so a minimal basis is formed by
#p_1,p_2,p_4# or #p_1,p_3,p_4#