21. \( \left(4 x^{3}+x^{2}+9 x+5\right) \div\left(x^{2}-1\right) \) \( \begin{array}{ll}\text { a. } 4 x+1+\frac{5 x+4}{x^{2}-1} & \text { c. } 4 x+1+\frac{13 x}{x^{2}} \\ \text { b. } 4 x-3+\frac{12 x+5}{x^{2}-1} & \text { d. } 4 x+5+\frac{6 x+}{x^{2}}\end{array} \)

To perform the division \( \left(4 x^{3}+x^{2}+9 x+5\right) \div\left(x^{2}-1\right) \), we can use polynomial long division. 1. Divide the leading term: \(4x^{3} \div x^{2} = 4x\). 2. Multiply \(4x\) by \(x^{2} - 1\): \(4x(x^{2} - 1) = 4x^{3} - 4x\). 3. Subtract: \((4x^{3} + x^{2} + 9x + 5) - (4x^{3} - 4x) = x^{2} + 13x + 5\). 4. Now, divide \(x^{2}\) by \(x^{2}\): \(1\). 5. Multiply: \(1 \cdot (x^{2} - 1) = x^{2} - 1\). 6. Subtract again: \((x^{2} + 13x + 5) - (x^{2} - 1) = 13x + 6\). At this point, we can write: \[ (4x + 1) + \frac{13x + 6}{x^{2} - 1}. \] Looking at the provided options, the correct answer is: **a.** \(4x + 1 + \frac{13x + 6}{x^{2} - 1}\).