1. **State the problem:** We have a quadratic equation $$x^2 + mx + 58 - m = 0$$ where $m$ is a real number. The roots of this equation are integers. We need to find the largest possible root. 2. **Recall the relationships between roots and coefficients:** For a quadratic equation $$x^2 + bx + c = 0$$ with roots $\alpha$ and $\beta$, we have: $$\alpha + \beta = -b$$ $$\alpha \beta = c$$ In our equation, $b = m$ and $c = 58 - m$, so: $$\alpha + \beta = -m$$ $$\alpha \beta = 58 - m$$ 3. **Use the given relation:** $$\alpha \beta - \alpha - \beta = 58$$ Substitute $\alpha + \beta = -m$ and $\alpha \beta = 58 - m$: $$ (58 - m) - (-m) = 58 $$ Simplify: $$ 58 - m + m = 58 $$ $$ 58 = 58 $$ This confirms the relation is consistent. 4. **Express $m$ in terms of roots:** From $\alpha + \beta = -m$, we get: $$ m = - (\alpha + \beta) $$ From $\alpha \beta = 58 - m$, substitute $m$: $$ \alpha \beta = 58 - (- (\alpha + \beta)) = 58 + \alpha + \beta $$ Rearranged: $$ \alpha \beta - \alpha - \beta = 58 $$ This matches the given relation. 5. **Rewrite the key equation:** $$ \alpha \beta - \alpha - \beta = 58 $$ Add 1 to both sides: $$ \alpha \beta - \alpha - \beta + 1 = 59 $$ Factor the left side: $$ (\alpha - 1)(\beta - 1) = 59 $$ 6. **Analyze integer factors:** Since $\alpha$ and $\beta$ are integers, $\alpha - 1$ and $\beta - 1$ are integers whose product is 59. 7. **Find factor pairs of 59:** 59 is prime, so the integer factor pairs of 59 are: $$ (1, 59), (59, 1), (-1, -59), (-59, -1) $$ 8. **Find corresponding roots:** For each pair $(p, q) = (\alpha - 1, \beta - 1)$: - If $(p, q) = (1, 59)$, then $\alpha = 2$, $\beta = 60$ - If $(p, q) = (59, 1)$, then $\alpha = 60$, $\beta = 2$ - If $(p, q) = (-1, -59)$, then $\alpha = 0$, $\beta = -58$ - If $(p, q) = (-59, -1)$, then $\alpha = -58$, $\beta = 0$ 9. **Find $m$ for each pair:** Recall $m = - (\alpha + \beta)$ - For $(2, 60)$: $m = - (2 + 60) = -62$ - For $(60, 2)$: $m = - (60 + 2) = -62$ - For $(0, -58)$: $m = - (0 - 58) = 58$ - For $(-58, 0)$: $m = - (-58 + 0) = 58$ 10. **Check roots for each $m$:** The roots are integers as required. 11. **Find the largest possible root:** Among all roots $2, 60, 0, -58$, the largest is $60$. **Final answer:** $$\boxed{60}$$