Theory: To find remainder of a number by 10 we just need to find the unit's digit of that number and the unit's digit is the remainder of the number by 10.

Let's take an example to understand the theory.
Remainder of 5 by 10 is 5, which is nothing but the unit's digit.
Remainder of 15 by 10 is also 5, which is nothing but the unit's digit. and so on.

So, to find the remainder when \(3^{4x+2}\) is divided by 10 we need to find the unit's digit of \(3^{4x+2}\)

Theory: To find the unit's digit of power of a number (let say n) we need to find the cycle of power of n and use that cycle to find the unit's digit of power of the exponent

Finding unit's digit of \(3^{4x+2}\)

Let's do that for 3 now
Unit's digit of \(3^1\) = 3
Unit's digit of \(3^2\) = 9
Unit's digit of \(3^3\) = 7 [ Just multiple unit's digit with 3 to get 9*3 = 7]
Unit's digit of \(3^4\) = 1 [ Just multiple unit's digit with 3 to get 7*3 = 1]
Unit's digit of \(3^5\) = 3 [ Just multiple unit's digit with 3 to get 1*3 = 3]
Unit's digit of \(3^6\) = 9 [ Just multiple unit's digit with 3 to get 3*3 = 9]

So, the cycle of power of 3 is 4. That means that \(3^5\) will have the same unit's digit as \(3^1\)
So, to find unit's digit of any power of 3 we need to divide the power of 3 by 4 and get the remainder(r)
Unit's digit of power of 3 = unit's digit of \(3^r\) then

So, lets divide 4x+2 by 4. We will get 2 as remainder.
=> \(3^{4x+2}\) will have the same remainder as \(3^2\) which is 9
=> remainder when \(3^{4x+2}\) is divided by 10 = Unit's digit of \(3^{4x+2}\) = 9

So, answer will be E
Hope it helps!

To Learn More about Remainders with 2,3,5,9,10