If x = 2 , we have x2 + 2x = 8 ( scrap because 8 isn't a prime number)

If x = 3 , we have x2 + 2x = 17 ( satisfy )

If x > 3 and x is a prime number then x =3k + 1 or 3k + 2 ( k is a natural number)

+) x = 3k +1

If x = 3k +1 then x2 divide to 3 has the remainder 1

and 2x divide to 3 has the remainder 2

=> 2x + x2 divided to 3 

But 2x + x2 > 3 

So 2x + x2 isn't a prime number

Do the similar to x = 3k+2

The answer is x = 3

If x = 2 , we have x² + 2^x = 8 ( scrap because 8 isn't a prime number)

If x = 3 , we have x² + 2^x = 17 ( satisfy )

If x > 3 and x is a prime number then x =3k + 1 or 3k + 2 ( k is a natural number)

+) x = 3k +1

If x = 3k +1 then x² divide to 3 has the remainder 1

and 2^x divide to 3 has the remainder 2

=> 2^x + x² divided to 3 

But 2^x + x² > 3 

So 2^x + x² isn't a prime number

Do the similar to x = 3k+2

The answer is x = 3

If x = 2 , we have x2 + 2x = 8 ( scrap because 8 isn't a prime number)

If x = 3 , we have x2 + 2x = 17 ( satisfy )

If x > 3 and x is a prime number then x =3k + 1 or 3k + 2 ( k is a natural number)

+) x = 3k +1

If x = 3k +1 then x2 divide to 3 has the remainder 1

and 2x divide to 3 has the remainder 2

=> 2x + x2 divided to 3 

But 2x + x2 > 3 

So 2x + x2 isn't a prime number

Do the similar to x = 3k+2

The answer is x = 3

If x = 2 , we have x2 + 2x = 8 ( scrap because 8 isn't a prime number)

If x = 3 , we have x2 + 2x = 17 ( satisfy )

If x > 3 and x is a prime number then x =3k + 1 or 3k + 2 ( k is a natural number)

+) x = 3k +1

If x = 3k +1 then x2 divide to 3 has the remainder 1

and 2x divide to 3 has the remainder 2

=> 2x + x2 divided to 3 

But 2x + x2 > 3 

So 2x + x2 isn't a prime number

Do the similar to x = 3k+2

The answer is x = 3