`x^4 + 11x^2 - 80` is a trinomial and can be factored by the method of splitting the middle term.

Find two number whose product is -80 and whose sum is 11. That is,

|__| * |__| = -80

|__| + |__| = 11

The two numbers are 16 and -5 because 16 times -5 is -80 and 16 - 5 is 11. Now rewrite the expression and replace `11x^2` with `16^2 - 5x^2` to get

`x^4 + 16x^2 - 5x^2 - 80`

Factor the expression further by grouping the four terms into two groups. The first group contains the first to terms and the second group contains the next two terms,

`(x^4 + 16x^2) - (5x^2 - 80)`

Factor out `x^2` from the first group and -5 from the second group,

`x^2(x^2 + 16) - 5(x^2 + 16)`

Factor out `x^2 + 16` from the expression,

`(x^2 + 16)(x^2 - 5)`

Thus the expression `x^4 + 11x^2 - 80` is completely factored `(x^2 + 16)(x^2 - 5)`

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