i don't really like to work in the i-j-k notation, so let's write

p = <2,3,4>

q = <1,3,2>

r = <2,1,a>

i suppose your dot means the dot product

Now, p.(q + r) = <2,3,4>.(<1,3,2> + <2,1,a>)

....................= <2,3,4>.<3,4,2+a>

....................= 6 + 12 + 8 + 4a

....................= 26 + 4a

Now, q.(p + r) = <1,3,2>.(<2,3,4> + <2,1,a>)

....................= <1,3,2>.<4,4,4+a>

....................= 4 + 12 + 8 + 2a

....................= 24 + 2a

Now we are told that p.(q+r) is 6 greater than q.(p+r), that is,

q.(p + r) + 6 = p.(q + r)

=> 24 + 2a + 6 = 26 + 4a

=> 2a = 4

=> a = 2