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When factoring, always start with the Greatest Common Factor, GCF (unless it is a one). The GCF here is 3:
And, similar to reducing fractions, keep factoring until you cannot factor any further. There's not much we can do to factor the 3. But we can try to factor the other factor. Since the first and last terms of are not perfect squares this expression will not fit any of the factoring patterns. But it is a trinomial which can be factored. Since there is really only one way to factor we know that the first terms will be 7a and a:
3(7a )(a )
Since 90 has 6 pairs of factors (1 and 90, 2 and 45, 3 and 30, 5 and 18, 6 and 15 and 9 and 10) we will have to figure out which pair works, if any. And since the 90 is negative there will be a "+" in one factor and a "-" in the other but we do not know which factor will have the "+" and which will have the "-".
From here can do one of two things:
DO a lot of trial and error. Try different factor pairs and try putting the "+" and "-" in the different factors. This process can be sped up if you keep in mind that one of the factors of 90 will be multiplied by the 7 in 7a and the other will be multiplied by just 1. And keep in mind that one of these products will be positive and one of them negative. And finally the positive and negative products will add up to -53x. So the negative factor must be "bigger" than the positive one by 53! All this leads me to think that the + and - will be located:
3(7a + )(a - )
Trying the various pairs of factor of 90 we should find that only one fits:
3(7a + 10)(a - 9)
The Quadratic formula can be used:
which simplifies down to:
This is short for: or
which simplify to: or
From these we get factors of:
(a - 9) and (7a - (-10)) (Note where the numerator and denominator went!)
or
(a - 9) and (7a + 10)
Either way the fully factored expression is:
3(7a + 10)(a - 9)
(