1. **State the problem:** Factor the quadratic expression $x^2 - 24x - 63$ into the form $(x - a)(x - b)$. 2. **Recall the factoring formula:** For a quadratic $x^2 + bx + c$, factors are $(x - m)(x - n)$ where $m$ and $n$ satisfy $m + n = -b$ and $mn = c$. 3. **Apply to our problem:** Here, $b = -24$ and $c = -63$. So, we need two numbers $m$ and $n$ such that $$m + n = 24$$ $$mn = -63$$ 4. **Find factors of -63 that sum to 24:** The pairs of factors of 63 are (1,63), (3,21), (7,9). Considering signs to get sum 24, try $-3$ and $21$: $-3 + 21 = 18$ (no), try $3$ and $-21$: $3 - 21 = -18$ (no), try $-7$ and $9$: $-7 + 9 = 2$ (no), try $7$ and $-9$: $7 - 9 = -2$ (no). Try $-1$ and $63$: $-1 + 63 = 62$ (no), try $1$ and $-63$: $1 - 63 = -62$ (no). Try $-9$ and $7$: $-9 + 7 = -2$ (no). Try $-21$ and $3$: $-21 + 3 = -18$ (no). Try $-63$ and $1$: $-63 + 1 = -62$ (no). Try $-24$ and $1$: product is not 63. 5. **Re-examine the sum condition:** The sum should be $-b = 24$, but original $b$ is $-24$, so $m + n = 24$. Since $c = -63$, one factor is positive and the other negative. Try $27$ and $-3$: $27 + (-3) = 24$ and $27 imes (-3) = -81$ (no). Try $21$ and $-3$: $21 + (-3) = 18$ (no). Try $-27$ and $3$: $-27 + 3 = -24$ (no). Try $-21$ and $3$: $-21 + 3 = -18$ (no). Try $-9$ and $7$: $-9 + 7 = -2$ (no). Try $9$ and $-7$: $9 + (-7) = 2$ (no). Try $-63$ and $1$: $-63 + 1 = -62$ (no). Try $63$ and $-1$: $63 + (-1) = 62$ (no). 6. **Use quadratic formula to find roots:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{24 \pm \sqrt{(-24)^2 - 4 \times 1 \times (-63)}}{2} = \frac{24 \pm \sqrt{576 + 252}}{2} = \frac{24 \pm \sqrt{828}}{2}$$ 7. **Simplify the square root:** $$\sqrt{828} = \sqrt{4 \times 207} = 2\sqrt{207}$$ 8. **Write roots:** $$x = \frac{24 \pm 2\sqrt{207}}{2} = 12 \pm \sqrt{207}$$ 9. **Final factorization:** $$x^2 - 24x - 63 = (x - (12 + \sqrt{207}))(x - (12 - \sqrt{207}))$$ This is the exact factorization over real numbers.