You don't need to find all the values

The solution to this problem is:

Step 1: $100^{x^2-6x+1}+5=10$

Step 2: $100^{x^2-6x+1}=10-5$

Step 3: $100^{x^2-6x+1}=5$

Step 4: $\ln(100^{x^2-6x+1})=\ln(5)$

Step 5: $(x^2-6x+1)\times\ln(100)=\ln(5)$

Step 6: $\displaystyle x^2-6x+1=\frac{\ln(5)}{\ln(100)}$

Step 7: Solve the quadratic equation to find the values of $x$ which will be: $x=5.88954754282$ and $x=0.110452457185$

Step 8: Add $5.88954754282$ and $0.110452457185$

Step 9: The answer will be $6$

WRITE THESE STEPS IF YOU GOT WRONG TO HAVE A CLEARER UNDERSTANDING.

THANK YOU.

$100^{x^2-6x+1} +5 = 10\\ 100^{x^2-6x+1} = 5\\ \log_{10}(10^2)^{x^2-6x+1} = \log_{10} 5\\ 2(x^2-6x+1) = \log_{10} 5\\ x^2 - 6x + 1 = \dfrac{\log_{10} 5}{2}\\ x^2 - 6x + 1 - \dfrac{\log_{10} 5}{2} = 0$

Given that $x^2 - (\text{Sum of roots})x + (\text{Product of roots}) = 0$

Sum of roots = $\boxed{6}$

$100^{x^2-6x+1} = 5$ , which, when take log 100, gives $x^2-6x+1 = \log \sqrt{5}$ , or simply $(x-3)^2 = 8+\log \sqrt{5}$ Hence, $x = 3\pm\sqrt{8+\log \sqrt{5}}$ , and their sum is 6.