Question 621122
I see two ways to the solution.
FACTORING:
If the numbers must be integers, the only solutions are included in the following products:
{{{35=7*5}}} or {{{35=(-7)*(-5)}}} and
{{{35=35*1}}} or {{{35=(-35)*(-1)}}}
The only one of those product whose factors add up to -12 is the product of
{{{highlight(-7)}}} and {{{highlight(-5)}}} .
USING QUADRATIC EQUATION:
The two solutions to the equation {{{x^2+12x+35=0}}} add up to -12 and multiply to + 35.
(The solutions to a quadratic equation add up to the opposite of the term in x, and their product is the independent term).
The quadratic formula says that the solutions are given by
{{{x = (-12 +- sqrt( 12^2-4*1*35 ))/(2*1)=(-12 +- sqrt( 144-140 ))/2=(-12 +- sqrt( 4))/2=(-12 +- 2)/2}}}
resulting in 
{{{x=(-12+2)/2=-10/2=-5}}} and {{{x=(-12-2)/2=-14/2=-7}}}