1. **State the problem:** Simplify or analyze the quadratic expression $x^2 + 42x + 13$. 2. **Formula and rules:** To factor a quadratic expression of the form $ax^2 + bx + c$, we look for two numbers that multiply to $ac$ and add to $b$. 3. **Calculate product and sum:** Here, $a=1$, $b=42$, and $c=13$. We need two numbers that multiply to $1 \times 13 = 13$ and add to $42$. 4. **Check factors of 13:** The factors of 13 are 1 and 13, which add to 14, not 42. Since no integer factors satisfy this, the quadratic cannot be factored easily over the integers. 5. **Use quadratic formula to find roots:** The roots are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-42 \pm \sqrt{42^2 - 4 \times 1 \times 13}}{2}$$ 6. **Calculate discriminant:** $$42^2 - 4 \times 1 \times 13 = 1764 - 52 = 1712$$ 7. **Simplify roots:** $$x = \frac{-42 \pm \sqrt{1712}}{2} = \frac{-42 \pm 2\sqrt{428}}{2} = -21 \pm \sqrt{428}$$ 8. **Final answer:** The quadratic expression $x^2 + 42x + 13$ cannot be factored nicely over integers, and its roots are $$x = -21 + \sqrt{428} \quad \text{and} \quad x = -21 - \sqrt{428}$$