m: X^3 +2*X^2 + 2*X + 3;
f: 2*X^2 + 2*X + 1;

q0: 4*X+4;
r0: 4*X+6;
expand(q0*f+r0);

q1: 4*X+5;
r1: 6;
expand(q1*r0+r1);

q2: 3*X+1;
r2: 0;
expand(q2*r1+r2);

Q0: matrix([ 0, 1 ], [ 1, (7*X+7)-q0 ]);
Q1: matrix([ 0, 1 ], [ 1, (7*X+7)-q1 ]);
Q2: matrix([ 0, 1 ], [ 1, (7*X+7)-q2 ]);

Q: expand(Q1 . Q0);
s: Q[1,1];
t: Q[1,2];
expand(s*m+t*f);

Q: expand(Q2 . Q);

s: Q[1,1];
t: Q[1,2];

expand(s*m+t*f);