1. **State the problem:** Divide the polynomial $k^2 - 45$ by $k - 7$ and express the result including any remainder as a simplified fraction. 2. **Recall the division formula:** For polynomials, dividing $f(k)$ by $g(k)$ gives quotient $q(k)$ and remainder $r(k)$ such that: $$f(k) = g(k) \times q(k) + r(k)$$ where the degree of $r(k)$ is less than the degree of $g(k)$. 3. **Set up the division:** Divide $k^2 - 45$ by $k - 7$ using polynomial long division. 4. **Divide the leading terms:** $\frac{k^2}{k} = k$. 5. **Multiply and subtract:** $$k \times (k - 7) = k^2 - 7k$$ Subtract this from $k^2 - 45$: $$\cancel{k^2} - 45 - (\cancel{k^2} - 7k) = 7k - 45$$ 6. **Divide the new leading term:** $\frac{7k}{k} = 7$. 7. **Multiply and subtract:** $$7 \times (k - 7) = 7k - 49$$ Subtract this from $7k - 45$: $$\cancel{7k} - 45 - (\cancel{7k} - 49) = 4$$ 8. **Remainder:** The remainder is $4$, which is a constant and degree less than divisor $k - 7$. 9. **Write the final answer:** $$\frac{k^2 - 45}{k - 7} = k + 7 + \frac{4}{k - 7}$$ **Final answer:** $k + 7 + \frac{4}{k - 7}$