How polynomial interpolation of polynomial multiplication algorithm works?
For testing, you can use SageMath very easily;
you had erros on
$$A(x_0) = 0, A(x_1) = 16, A(x_2) = 85, A(x_3)=208, \\B(x_0) = -1, \color{red}{B(x_1) = -2}, \color{red}{B(x_2) = -5},\color{red}{B(x_3) = 8}$$
    R = PolynomialRing(QQ, 'x')
    A = 3 *x^2-4*x+1
    B = x -2
    a0 = A(1)
    a1 = A(3)
    a2 = A(6)
    a3 = A(9)
    b0 = B(1)
    b1 = B(3)
    b2 = B(6)
    b3 = B(9)
    c0 = a0*b0
    c1 = a1*b1
    c2 = a2*b2
    c3 = a3*b3
    
    print( a0,a1,a2,a3)
    print( b0,b1,b2,b3)
    print( c0,c1,c2,c3)
    
    f = R.lagrange_polynomial([(1,c0),(3,c1),(6,c2),(9,c3)])
    f

Outputs
0 16 85 208
-1 1 4 7
0 16 340 1456
3*x^3 - 10*x^2 + 9*x - 2