===========================================================
Explanation:
Cut the x coefficient (10) in half to get 10/2 = 5. Then square this to get 5^2 = 25.
We'll add 1 to both sides so that the "24" turns into "25", thereby completing the square
x^2 + 10x + 24 = 0
x^2 + 10x + 24+1 = 0+1
x^2 + 10x + 25 = 1
Notice on the left hand side we have something of the form A^2+2AB+B^2 where A = x and B = 5. We can factor this into (A+B)^2, which is the whole reason why we completed the square. You can use the FOIL rule to see how (A+B)^2 expands out into A^2+2AB+B^2. Factoring reverses this process.
This means x^2+10x+25 factors to (x+5)^2 and we now have these steps
(x+5)^2 = 1
x+5 = sqrt(1) or x+5 = -sqrt(1)
x+5 = 1 or x+5 = -1
x = 1-5 or x = -1-5
x = -4 or x = -6 are the two solutions
------------------
Let's check x = -4 to see if it works or not
x^2 + 10x + 24 = 0
(-4)^2 + 10(-4) + 24 = 0
16 - 40 + 24 = 0
-24 + 24 = 0
0 = 0
We get a true equation. That confirms x = -4 is a solution.
If we tried x = -6, then,
x^2 + 10x + 24 = 0
(-6)^2 + 10(-6) + 24 = 0
36 - 60 + 24 = 0
-24 + 24 = 0
0 = 0
That x value is confirmed as well.