Question

How do you factor `2x^2-5x-3`?

Answer 1

`(x-3) (2x+1)`

Explanation:

Factor by grouping
Factor `2x^2−5x−3`

`2x^2−5x−3`

`=(x-3) (2x+1)`

Answer 2

`(x-3)(2x+1) color(white)("d") harr color(white)("d") (2x+1)(x-3)`

Explanation:

These are not always straight forward but lets have a 'play' and see what we get.

Consider whole number factors:

The only whole number factors of 2 are `1 and 2`
The only whole number factors of 3 are `1 and 3`

Structure test 1:

`(x+?)(2x+?) ->(x+3)(2x+1) -> color(red)("Fail")`
We need negative values and the above will only give positive

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Structure test 2:
`(x+-?)(2x+-?)->(x+3)(2x-1) ->2x^2-x+6x-3`

`color(white)("dddddddddddddddddddddddddddd")->2x^2color(red)(+5x)-3 ->color(red)("Fail")`
We need `-5x`

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Structure test 3:
`(x+-?)(2x+-?)->(x-3)(2x+1) ->2x^2+x-6x-3`
`color(white)("ddddddddddddddddddddddddddd")->2x^2-5x-3 ->color(green)("Works")`

After some practice you start to recognise what is needed reducing the amount of work. On the other hand some of them are blighters to figure out.

It is all down to PRACTICE!