Explanation:
we are looking for an expression like
(ay + b)(cy + d)
so that when doing all the multiplications this is equal to our original expression.
that is called factoring, as we are finding the 2 terms that when multiplied with each other create the original expression. we are finding the factors.
and we are normally looking for integer values for a, b, c, d.
ay×cy + b×cy + d×ay + b×d
acy² + (bc + da)y + b×d = 16y² + 62y - 45
so, we know that one of b, d must be negative, the other positive.
and a, c must have the same sign, so that their product is positive.
anyway, we compare now
ac = 16
bc + da = 62
bd = -45
the possible factors of 45 are (remember the different signs)
1 45
3 15
5 9
the possible factors of 16 are (remember they need the same signs)
1 16
2 8
4 4
let's start with the assumption a=4, c=4
is there a fitting factor pair of 45 to allow us to get
bc + da = 62 ?
b×4 + d×4 = 62
b + d = 62/4 = 15.5
since b and d should be integer, we cannot get a non-integer as their sum. so, that does not work.
assume a = 2, c = 8
b×8 + d×2 = 62
4b + d = 31
and here we find with a little trial and error with the factor pairs of 45 that
b = 9 and d = -5 fit that criteria.
4×9 - 5 = 36 - 5 = 31
perfect.
so, our factoring is
(2y + 9)(8y - 5)
control :
2y×8y + 2×-5y + 9×8y + 9×-5 =
= 16y² -10y + 72y - 45 = 16y² + 62y - 45
correct.