x is the product of 4 consecutive positive integers, that means its even number (because there will be 2 even numbers and any positive integer multiplied by even gives even)

This makes 'A' is True. i.e. n is odd as even + 1 is odd

for 'B' lets put some values.
it is given that x is product of 4 consecutive positive integers.
Assume consecutive numbers are 1,2,3, and 4 therefore product will be 24.

n = 24 + 1 = 25 (We proved that n is not prime)

For 'C' we cau put values and find or we can use algebra 
(It's easy to find through values)
Using algebra
Let x = y(y + 1)(y + 2)(y + 3)

x = y⁴ + 6y³ + 11y² + 6y
Adding and subtracting 1

x = y⁴ + 6y³ + 11y² + 6y + 1 - 1
x = (y² + 3y + 1)² - 1

n = x + 1
n = (y² + 3y + 1)² - 1 + 1
n = (y² + 3y + 1)²

n is perfect square

Therefore option 1. A and C only