Question 115382
Let one of the integers be p and the other be q.


The sum of two integers is -8:  {{{p+q=-8}}}


Their product is -20:  {{{pq=-20}}}


{{{p+q=-8}}} => {{{p=-8-q}}}


Substituting:
{{{(-8-q)q=-20}}}


Distribute:
{{{-8q-q^2=-20}}}


Multiply by -1, Add -20 to both sides, arrange in standard form:
{{{q^2+8q-20=0}}}


{{{10*(-2)=-20}}} and {{{10 + (-2)=8}}}, so:


{{{q^2+8q-20=(q+10)(q-2)=0}}}


{{{q = -10}}} or {{{q = 2}}}


Since {{{pq=-20}}} if {{{q=-10}}} then {{{p=2}}} and if {{{q=2}}} then {{{p=-10}}}, so your integers are -10 and 2.


Check:

{{{-10+2=-8}}}
{{{(-10)*2=-20}}}