Okay, if [tex](x+5)[/tex] is a factor of [tex]x^2+8x+15[/tex] then we know that [tex]x^2+8x+15 = (x+5)( \cdots)[/tex] for some other factor which we don't know but want to find out.
[tex]x^2[/tex] is the highest power on the left side of the equation so we know we want something in the missing bracket that when we times it by x we get x². We therefore know that we need an x in the missing bracket and so it looks like this [tex]x^2+8x+15 = (x+5)(x + \cdots)[/tex]. When we expand that out we get the x² term (by multiplying the two x's) but also we get a +5x (from multiplying the 5 and the x) but by comparing it to the left hand side again, we see we want it to be +8x, we are +3x short. How can we get +3x?
If we add 3 to the missing bracket we get [tex]x^2+8x+15 = (x+5)(x + 3)[/tex] so we have the [tex]x^2+5x + \cdots[/tex] from the first expansion and now we have [tex]\cdots + 3x + 15[/tex] from the second expansion. We have obtained the extra 3x we need to get to +8x but it has also added on 15 (by multiplying 3 and 5). Luckily, this is also part of the left hand side and sothe two sides match and we have found the missing factor: [tex](x+3).[/tex]
