1. **State the problem:** Divide the polynomial $x^2+4$ by $x-3$ and express the result in the form $$p(x)+\frac{k}{x-3}$$ where $p(x)$ is a polynomial and $k$ is an integer. 2. **Recall the division formula:** When dividing polynomials, $$\frac{f(x)}{g(x)} = q(x) + \frac{r(x)}{g(x)}$$ where $q(x)$ is the quotient polynomial and $r(x)$ is the remainder polynomial with degree less than $g(x)$. 3. **Perform polynomial division:** Divide $x^2 + 4$ by $x - 3$. - Divide the leading term $x^2$ by $x$ to get $x$. - Multiply $x$ by $x - 3$ to get $x^2 - 3x$. - Subtract this from $x^2 + 4$: $$ (x^2 + 4) - (x^2 - 3x) = 3x + 4 $$ - Now divide $3x$ by $x$ to get $3$. - Multiply $3$ by $x - 3$ to get $3x - 9$. - Subtract this from $3x + 4$: $$ (3x + 4) - (3x - 9) = 13 $$ 4. **Write the quotient and remainder:** Quotient polynomial is $x + 3$, remainder is $13$. 5. **Express the final answer:** $$\frac{x^2 + 4}{x - 3} = x + 3 + \frac{13}{x - 3}$$ This matches the required form with $p(x) = x + 3$ and $k = 13$.