We can do this in two different ways:
Method 1
Since
\((x + 5)\)
is a factor, we can use it to obtain the other factor by long division(see attachment).
\({x}^{2} + 8x + 15 = (x + 5)(x + 3)\)
Hence the other factor is
\((x + 3)\)
Method 2
We can factor the trinomial to obtain the other factor.
Two factors of 15 that add up to 8 are,
\(5,3\)
We split the middle term as follows:
\({x}^{2} + 8x + 15 = {x}^{2} + 3x + 5x + 15\)
\({x}^{2} + 8x + 15 = x(x + 3) + 5(x + 3)\)
\({x}^{2} + 8x + 15 = (x + 3) (x + 5)\)
Hence the other factor is
\((x + 3)\)
