Follow the steps for graphing a rational function to graph the function \( P(x)=\frac{x^{4}+x^{2}+3}{x^{2}-81} \) If needed, first write the given function as a single rational expression. Then, factor the numerator and de of \( P(x) \). Select the correct choiee and, if necessary, fill in the answer box to complete your choice. \( P(x)=\square \) (Type your answer in factored form. Do not simplify.)

To graph the rational function \( P(x) = \frac{x^{4} + x^{2} + 3}{x^{2} - 81} \), we first need to factor both the numerator and the denominator. 1. **Factor the denominator**: The denominator \( x^{2} - 81 \) is a difference of squares, which can be factored as: \[ x^{2} - 81 = (x - 9)(x + 9) \] 2. **Factor the numerator**: The numerator \( x^{4} + x^{2} + 3 \) is a quadratic in terms of \( x^{2} \). We can let \( y = x^{2} \), so we rewrite the numerator as: \[ y^{2} + y + 3 \] To factor \( y^{2} + y + 3 \), we can use the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 - 12}}{2} = \frac{-1 \pm \sqrt{-11}}{2} \] Since the discriminant is negative (\(-11\)), the numerator does not factor over the real numbers. Therefore, we leave it in its original form. 3. **Write the function as a single rational expression**: The function \( P(x) \) can be expressed as: \[ P(x) = \frac{x^{4} + x^{2} + 3}{(x - 9)(x + 9)} \] Thus, the final factored form of \( P(x) \) is: \[ P(x) = \frac{x^{4} + x^{2} + 3}{(x - 9)(x + 9)} \] In the answer box, you would write: \[ P(x) = \frac{x^{4} + x^{2} + 3}{(x - 9)(x + 9)} \]