Polynomial

g (x) =  f (x + 7) =    (x + 7)^7  - 3(x + 7)^3  + 2    =

1x^7 + 49 x^6 + 1029 x^5 + 12005 x^4 + 84032 x^3 + 352884 x^2 + 823102 x + 822516

Just add the   red integers to get your answer

Alternative solution:

The sum of coefficients of a polynomial $p(x)$ is actually just $p(1)$.

Proof: (You can omit this part if you just want the answer instead of the explanation.)

Let $p(x) = a_n x^n + a_{n - 1}x^{n - 1} + a_{n - 2}x^{n - 2} + \cdots + a_2 x^2 + a_1 x + a_0$.

Then $p(1) = a_n 1^n + a_{n - 1} 1^{n- 1} + \cdots + a_1 \cdot 1 + a_0 = a_n + a_{n -1} + a_{n-2} + \cdots + a_1 + a_0$, which is exactly the sum of coefficients of p(x).

Therefore, we just find $g(1) = f(1 + 7) = f(8)$. The sum of coefficients of g(x) is $f(8) = 8^7 - 3\cdot 8^3 + 2$.