Suppose $f^{\prime}(x)=-7 x^{10}+4 x^{5}-7 x^{3}+8$. What particular solution that satisfies $f(-1)=-\frac{1247}{132} ?$

Step 1 ：Given the derivative of a function as \(f^{\prime}(x)=-7 x^{10}+4 x^{5}-7 x^{3}+8\)

Step 2 ：Integrate the derivative to find the general solution of the function, which is \(f(x) = C - \frac{7}{11}x^{11} + \frac{2}{3}x^{6} - \frac{7}{4}x^{4} + 8x\)

Step 3 ：Substitute the given point \((-1, -\frac{1247}{132})\) into the general solution to find the constant of integration, which gives the equation \(C + 1.0 = 0\)

Step 4 ：Solve the equation to find \(C = -1.0\)

Step 5 ：Substitute \(C = -1.0\) back into the general solution to find the particular solution, which is \(f(x) = -\frac{7}{11}x^{11} + \frac{2}{3}x^{6} - \frac{7}{4}x^{4} + 8x - 1\)

Step 6 ：\(\boxed{f(x) = -\frac{7}{11}x^{11} + \frac{2}{3}x^{6} - \frac{7}{4}x^{4} + 8x - 1}\) is the particular solution that satisfies \(f(-1)=-\frac{1247}{132}\)