Question 320341
If you graph the function, you will see the roots clearly. 
Unfortunately due to the limitations of this graphing utility, they don't all show up. 
{{{x=2}}}, double root
{{{x=4}}}
{{{x=5}}}
{{{x=7}}}, double root
{{{x=8}}}
Verify that they are indeed roots by plugging them into the equation.  
{{{ graph( 300, 300, -2,9, -4, 5,(x-8)*(x-7)*(x-2)*(x-4)*(x-5)) }}} 


{{{x^7 - 35x^6 + 507x^5 - 3921x^4 + 17388x^3 - 43956x^2 + 58352x - 31360 = 
(x-2)^2(x-4)(x-5)(x-7)^2(x-8)}}}