$$\left(x^2 - 3x + 2\right) \mid \left(x^3 - 6x^2 + Ax + B\right) \Rightarrow A^2 + B^2 =$$

we have given 

$$\left(x^2 - 3x + 2) $$ =0                             equation 1 

and $$ left(x^3 - 6x^2 + Ax + B) $$ = 0           equation 2 

now we have to factorize  equation 1 we get the equation 

(x - 1) (x - 2) = 0 

x = 1, 2 

put the value of x in equation 2 we get 

2a +  b = 16                                                       equation 3

a + b = -5                                                          equation 4 

on solving eq 3 and eq 4 we get   

a = 11

b = 6 

then a^2 + b^2 = 157 answer