Complete the parts below to completely factor $x^4 + 2x^2 + 81$.

1. To make a perfect square, what does the middle term need to be (instead of $2x^{2}$)?

   Hint (a):

   $x^{4} + \text{___} + 81 = (x^2 + 9)^2$

2. Explain how $x^{4} + 18x^{2} + 81 − 16x^{2}$ is equivalent to the original expression.

   Hint (b):

   What term was added and subtracted (for a total of $0$) to the original expression?

3. Factor $x^{4} + 18x^{2} + 81$.

4. Factor $u^{2} − 16x^{2}$.

   Hint (d):

   This is a difference of squares.

5. Let $u = x^{2} + 9$ and use your answers to parts (a) through (d) to factor the original expression completely.