We have
a^3+12a^2+49a + 69=0
=> (a+4)^3 + (a+5) = 0
or (a+4)^3 + (a+4) + 1= 0\cdots(1)
b^3 -9b^2+ 28b -31 = 0
=>(b-3)^3 + (b-4) = 0
=>(b-3)^3 + (b-3) -1 = 0\cdots(2)
if we define f(x) = x^3+x knowing that f(x) is odd function
(f(a+4) = -1 and -f(b-3) = 1
so (a+4) = - (b- 3)
or a + b = - 1
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