Let f(x) and g(x) be polynomials defined as f(x) = x^2 + x g'(1) +...

Given the functions:
$f(x) = x^2 + x g'(1) + g''(2)$
$g(x) = x^3 + x^2 f'(1) + x f''(2)$

Differentiating $f(x)$ with respect to $x$:
$f'(x) = 2x + g'(1)$
$f''(x) = 2$
Thus, $f''(2) = 2$.

Substitute $f''(2) = 2$ into $g(x)$:
$g(x) = x^3 + x^2 f'(1) + 2x$

Differentiating $g(x)$ with respect to $x$:
$g'(x) = 3x^2 + 2x f'(1) + 2$
$g''(x) = 6x + 2f'(1)$

Evaluate $g'(x)$ at $x = 1$ and $g''(x)$ at $x = 2$:
$g'(1) = 3(1)^2 + 2(1)f'(1) + 2 = 5 + 2f'(1)$
$g''(2) = 6(2) + 2f'(1) = 12 + 2f'(1)$

Now, evaluate $f'(x) = 2x + g'(1)$ at $x = 1$:
$f'(1) = 2(1) + g'(1)$
Substitute $g'(1) = 5 + 2f'(1)$ into this equation:
$f'(1) = 2 + 5 + 2f'(1)$
$f'(1) = 7 + 2f'(1)$
$\Rightarrow f'(1) = -7$

Using $f'(1) = -7$, we find $g'(1)$ and $g''(2)$:
$g'(1) = 5 + 2(-7) = -9$
$g''(2) = 12 + 2(-7) = -2$

Substitute these values back to construct $f(x)$ and $g(x)$:
$f(x) = x^2 - 9x - 2$
$g(x) = x^3 - 7x^2 + 2x$

Evaluate both functions at $x = 10$:
$f(10) = (10)^2 - 9(10) - 2 = 100 - 90 - 2 = 8$
$g(10) = (10)^3 - 7(10)^2 + 2(10) = 1000 - 700 + 20 = 320$

Finally, the sum is:
$f(10) + g(10) = 8 + 320 = 328$.

Answer: $328$