Induction.

First, we verify a "base case". (like n = 1) I'll assume n is a natural number.

base case: (n = 1)

3^(4+2) + 5^(2+1) = 3^7 + 5^3 = 854. (( 854/14 = 61. so we are good here)

induction step:

Suppose that P(n) is true. That is to say, suppose "3^(4n+2) + 5^(2n+1) is divisible by 14)" = "3^(4n+2) + 5^(2n+1) = 14k for some k" (1)

Show P(n+1) is true.

3^(4(n+1)+2) + 5^(2(n+1)+1)

= 3^(4n+6) + 5^(2n+3)

= 3^4 * 3^(4n+2) + 5^2 * 5^(2n+1)

substitute using (1)

= 81 * 3^(4n+2) + 25 * (14k - 3^(4n+2))

= 81 * 3^(4n+2) - 75 *3^(4n+2) + 25*14k

= 6 * 3^(4n+2) + 25*14k

...

I've made a mistake somewhere in my algebra, b/c 6*3^(4n+2) for n=1 is not divisible by 14. sorry I can't find it (sleepy) but you get the idea?