Question

How do you factor `7k^2+19k+10`?

Answer 1

Look for factor pairs of `7` and of `10` which when cross multiplied equal `19`.
This should allow you to find:
`color(white)("XXX")(k+2)(7k+5)`

Explanation:

Note that since all terms in the given expression are positive, we only need to consider positive factors.

Factors of `7`
`color(white)("XXX"){color(red)(color(white)("")(1,7))}`
Factors of `10`
`color(white)("XXX"){color(blue)(color(white)("")(1,10),(2,5))}`

Using `" X "` to indicate cross multiplication:
Since `color(red)(1)xxcolor(blue)(5)+color(red)(7)xxcolor(blue)(2) = 19`

`color(white)("XXX"){:(color(red)(1)),(color(red)(7)):}" X "{:(color(blue)(2)),(color(blue)(5)):}=19`

The required factors are
`color(white)("XXX")(color(red)(1)k+color(blue)(2))(color(red)(7)k+color(blue)(5))`

Note that this method only works since the coefficients are integer.