For how many natural numbers x, is the expression: (x ² + 2x + 3) divisible by 35 ?

For any natural number x, (x^2+2x+3) mod 5 is congruent to 2, 3 or 4. This means the value of x^2+2x+3 can never be a multiple of 5. Then the answer to the problem is

*there are no natural numbers x which make x^2+2x+3 a multiple of 35*.