Question

1. ( n^2 - n - 90 )

Explanation:

Step1: Find two numbers

We need two numbers that multiply to $-90$ and add up to $-1$.
Let the numbers be $a$ and $b$, so $a\times b=-90$ and $a + b=-1$.
After checking, we find that $a = -10$ and $b = 9$ since $(-10)\times9=-90$ and $-10 + 9=-1$.

Step2: Factor the quadratic

Using the numbers from Step 1, we can factor $n^{2}-n - 90$ as:
$n^{2}-n - 90=n^{2}-10n+9n - 90$
Group the terms:
$=(n^{2}-10n)+(9n - 90)$
Factor out the common factors from each group:
$=n(n - 10)+9(n - 10)$
Now, factor out the common binomial factor $(n - 10)$:
$=(n - 10)(n + 9)$

Answer:

$\boxed{(n - 10)(n + 9)}$