Answer with Step-by-step explanation:
Let Z be a set of integers 
We have to prove that  is odd for all value of x
 is odd for all value of x  Z.
Z.
There are two cases 
1.When x is an odd integer number 
2.when x is even integer number 
1.When x is an odd integer  number 
When x is positive odd integer 
Then  is positive odd number (because square of odd positive number is always positive an odd number )
  is positive odd number (because square of odd positive number is always positive an odd number )
5x is also odd number 
-5x -1 = -even number 
Suppose x=3
Then -15-1=-16
Odd number - even number=Odd number 
Hence , is an odd number .
 is an odd number .
When x is negative odd number 
Then  x square is positive odd number  and 5x is negative odd term term
Therefore,odd number + odd number -1=Even  number -1=Odd number 
Hence  is an odd number.
 is an odd number.
2.When  x is an even number 
When x is positive even number 
x square is positive even number  and 5x is positive even number 
Even number -Even number -1 =Even number -odd number =Odd number 
Suppose x=4 
 =16-20-1=-5=Odd number
=16-20-1=-5=Odd number 
When x is negative even number
Then x square is positive even number  and 5x is negative even number 
 =Even number +Even number -1=Even number -1=Odd number
=Even number +Even number -1=Even number -1=Odd number 
Hence, for all elements of Z is odd.
is odd.