Question

How do you factor the expression `15m^2 + 20mp - 12mp - 16p^2`?

Answer 1

`15m^2+20mp-12mp-16p^2=(5m-4p)(3m+4p)`

Explanation:

Factor by grouping:

`15m^2+20mp-12mp-16p^2`

`=(15m^2+20mp)-(12mp+16p^2)`

`=5m(3m+4p)-4p(3m+4p)`

`=(5m-4p)(3m+4p)`

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Footnote

Note that the hard/fun part of the problem has already been done for you.

Normally you would be asked to factor something like:

`15m^2+8mp-16p^2`

It would then be up to you to find the right split of `8mp` as `20mp-12mp` to get the factoring by grouping to work.

To help with that, you could use an "AC method":

Ignoring signs, the three coefficients of the quadratic are:

`A=15`, `B=8` and `C=16`.

Noting that the sign on the last term is negative, you then look for a factorisation of `AC = 15*16 = 240` into a product of two numbers whose difference is `B=8`. If the sign on the last term was positive, then you would look for a pair of factors whose sum was `B`.

With a little thinking, experimentation, searching and/or trial and error, you can find the pair `20, 12` which satisfies the requirements: `20xx12 = 240` and `20-12 = 8`.

Then you would replace `8mp` with `20mp - 12mp` before factoring by grouping.