Find the Product:
\[ (2a – 3b – 2c) \] \[ (4a^2 + 9b^2 + 4c^2 + 6ab – 6bc + 4ca) \]
Solution:
Using identity:
\[ (a+b+c)(a^2+b^2+c^2-ab-bc-ca) = a^3+b^3+c^3-3abc \]
Rewrite:
\[ 2a-3b-2c = 2a+(-3b)+(-2c) \]
So, \[ a=2a,\qquad b=-3b,\qquad c=-2c \]
\[ (2a – 3b – 2c) \] \[ (4a^2 + 9b^2 + 4c^2 + 6ab – 6bc + 4ca) \]
\[ = (2a)^3+(-3b)^3+(-2c)^3 -3(2a)(-3b)(-2c) \]
\[ = 8a^3-27b^3-8c^3-36abc \]