Sum of Real Roots of Polynomial Equation

Let $k = x^2 + 3x$. Then $f(x) = k(k+2) = k^2 + 2k$. $f(x) + 1 = k^2 + 2k + 1 = (k+1)^2$. $\sqrt{f(x) + 1} = |k+1| = 9701$. Since $k+1 = x^2 + 3x + 1$, we have $x^2 + 3x + 1 = 9701$ or $x^2 + 3x + 1 = -9701$.

1. $x^2 + 3x - 9700 = 0$: Discriminant $D = 9 + 4(9700) > 0$, real roots exist. Sum of roots = $-3/1 = -3$.
2. $x^2 + 3x + 9702 = 0$: Discriminant $D = 9 - 4(9702) < 0$, no real roots. Thus, the sum of all real roots is $-3$.