If [( a +2 b +3 c ) ( b +2 c +3 a )] ( c +2 a +3 b )=54 , then the value of [ array lll a & b & c array ] is

Step-by-step solution

R.H.S. of the given equality can be written as aligned &(2 a +3 b + c ) [( a +2 b +3 c ) (3 a + b +2 c )] &=(2 a +3 b + c ) [3( a a )+( a b )+2( a c ) &+6( b a )+2( b b )+4( b c ) &+9( c a )+3( c b )+6( c c )] &=(2 a +3 b + c )[0+( a b )+2( a c ) &-6( a b )+0+4( b c ) &=(2 a +3 b + c )[-5( a c )-3( b c )+( b c )-7( a c )] &=-10[ a ( a b )]+2[ a ( b c )]-14[ a ( a c )] &-15[ b ( a b )]+3[ b ( b c )]-21[ b ( a c )] &-5[ c ( a b )]+[ c ( b c )]-7[ c ( a c )] &= 0+2[ a b c ]+0 &+0+0 &=-5[ a b c ]+0+21[ a b c ] &= 18[ a