1. **State the problem:** Simplify or factor the quadratic expression $12m^2 + m - 90$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multiply to $a \times c$ and add to $b$. 3. **Calculate the product:** Here, $a = 12$, $b = 1$, and $c = -90$. So, $a \times c = 12 \times (-90) = -1080$. 4. **Find two numbers that multiply to $-1080$ and add to $1$:** These numbers are $36$ and $-30$ because $36 \times (-30) = -1080$ and $36 + (-30) = 6$ (not 1). Since 36 and -30 do not add to 1, try other factor pairs. Try $45$ and $-24$: $45 \times (-24) = -1080$ and $45 + (-24) = 21$ (no). Try $-36$ and $30$: $-36 \times 30 = -1080$ and $-36 + 30 = -6$ (no). Try $-45$ and $24$: $-45 \times 24 = -1080$ and $-45 + 24 = -21$ (no). Try $60$ and $-18$: $60 \times (-18) = -1080$ and $60 + (-18) = 42$ (no). Try $-60$ and $18$: $-60 + 18 = -42$ (no). Try $54$ and $-20$: $54 + (-20) = 34$ (no). Try $-54$ and $20$: $-54 + 20 = -34$ (no). Try $90$ and $-12$: $90 + (-12) = 78$ (no). Try $-90$ and $12$: $-90 + 12 = -78$ (no). Try $108$ and $-10$: $108 + (-10) = 98$ (no). Try $-108$ and $10$: $-108 + 10 = -98$ (no). Try $15$ and $-72$: $15 + (-72) = -57$ (no). Try $-15$ and $72$: $-15 + 72 = 57$ (no). Try $9$ and $-120$: $9 + (-120) = -111$ (no). Try $-9$ and $120$: $-9 + 120 = 111$ (no). Try $18$ and $-60$: $18 + (-60) = -42$ (no). Try $-18$ and $60$: $-18 + 60 = 42$ (no). Try $24$ and $-45$: $24 + (-45) = -21$ (no). Try $-24$ and $45$: $-24 + 45 = 21$ (no). Try $30$ and $-36$: $30 + (-36) = -6$ (no). Try $-30$ and $36$: $-30 + 36 = 6$ (no). Since no integer pairs add to 1, the quadratic is not factorable over integers. 5. **Use the quadratic formula:** $$m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1^2 - 4 \times 12 \times (-90)}}{2 \times 12} = \frac{-1 \pm \sqrt{1 + 4320}}{24} = \frac{-1 \pm \sqrt{4321}}{24}$$ 6. **Final answer:** The quadratic does not factor nicely, and the solutions are $$m = \frac{-1 + \sqrt{4321}}{24} \quad \text{or} \quad m = \frac{-1 - \sqrt{4321}}{24}$$