Question

How do you factor the trinomial `10x^2 + 11x + 3`?

Answer 1

`(2x+1)(5x+3)`

Explanation:

`"'split the x-term and factorise by grouping"`

`10x^2+5x+6x+3larr (5x+6x=11x)`

`color(red)(5x)(2x+1)color(red)(+3)(2x+1)`

`"factor out "(2x+1)`

`(2x+1)(color(red)(5x+3))`

`rArr10x^2+11x+3=(2x+1)(5x+3)`

Answer 2

`(5x+3)(2x+1)`

Explanation:

`10x^2 +11x+3`

Find factors of `10 and 3` whose products ADD up to `11`

`3` is a prime number, so there are not very many options to try.
Using `10` will not work - the sum will be too big, try `2 and 5`
Cross multiply the factors and add the products till you get `11`.

`" "10 and 3`
`" "darr" "darr`

`" "5color(white)(xxxx)3" "rarr 2xx3 = 6`
`" "2color(white)(xxxx)1" "rarr 5xx1 = ul5`
`color(white)(xxxxxxxxxxxxxxxx.xx)11`

The signs are both positive.

`(5x+3)(2x+1)`

There are clues in the original expression.

`10x^2 +11xcolor(blue)(+)3`

`color(blue)(+)` sign tells you 2 things:
`rarr` ADD the products of factors of `10 and 3`
`rarr` the SIGNS will be the SAME.

The (+) sign of the 'b' which is `+11` tells you they will be positive.