Question

How do solve `n^2 - 10n + 22 = -2`?

Answer 1

`n^2-10n+22= -2`
is equivalent to
`n^2-10n+24=0`

We would like to factor the left hand size to get something of the form
`(n+a)(n+b) = n^2-10n+24`

Since the final term is positive, `a` and `b` must have the same sign
and
since the coefficient of `x` is negative they must both be negative.

So we are looking for negative values `a` and `b` such that
`a+b = -10`
and
`ab = +24`

Assuming integer solutions there are very few possibilities.
By trial and error we can derive
`(n-6)(n-4) = n^2-10n+24 = 0`

which implies that
`n=6` or `n=4`