Completing the square
alrighty
minus 31 both sides
x²+12x=-31
take 1/2 of the linear coefient and square it
12 is the linear coefient
12/2=6, (6)²=36
add that to both sides
x²+12x+36=36-31
factor perfect square trinomial
(x+6)²=5
square root both sides, remember to take positive and negative root
x+6=+/-√5
so what you want to write is
x+6=√5 and
x+6=-√5
the solutions
minus 6 both sides
x=-6+√5 and -6-√5
A²=25 then a = + 5 & a =-5, so this ellipse intercepts the x axis at the points +5 & -5
I will take you the steps to obtain the slopes passing two points
step 1 : list the parameters

step 2: Apply the equation to obtain the slope between two points