1. **Problem:** Factorize the expression $x^2 - x - 90$ using the AC method. 2. **Formula and rules:** The AC method involves multiplying the coefficient of $x^2$ (which is $1$) by the constant term (which is $-90$), then finding two numbers that multiply to $-90$ and add to the coefficient of $x$ (which is $-1$). 3. **Step-by-step solution:** - Multiply $a$ and $c$: $1 \times (-90) = -90$ - Find two numbers that multiply to $-90$ and add to $-1$: these are $9$ and $-10$ because $9 \times (-10) = -90$ and $9 + (-10) = -1$ - Rewrite the middle term using these numbers: $$x^2 + 9x - 10x - 90$$ - Group terms: $$(x^2 + 9x) + (-10x - 90)$$ - Factor each group: $$x(x + 9) - 10(x + 9)$$ - Factor out the common binomial: $$(x - 10)(x + 9)$$ 4. **Final answer:** $$x^2 - x - 90 = (x - 10)(x + 9)$$