how do you factor this out?

I don't know if your teacher said it was okay to use square roots or not:

http://www.algebrahelp.com/lessons/equations/solvebyfactoring/pg3.htm
According to the site:

The "Solve by Factoring" process will require four major steps:

1. Move all terms to left of the =, and change their sign.
2. Factor the equation completely.
3. Set each factor equal to zero, and solve.
4. List each solution (from Step 3) as a solution to the original equation.

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Your problem already has everything "to one side".
In following these steps, your problem comes out like:

A: h^4 + 3h^2 -10 = 0
Setting it equal to zero, thus forming your "trinomial"

B: (h² + 5)(h² - 2) = 0
Factoring the trinomal. Don't even make me being to explain how I derived it. I just used the example set forth on the website. Alright, I'l explain. I used the middle number of the trinomial as a point of addition; addition that compared using the same numbers that would MULTIPLY into -10. Here, we see that:

1. 5 - 2 = 3
2. 5 • -2 = -10 (5 times -2..)
3. x^4 minus x² = x²

C: Set both equations equal to zero and solve:

(h² + 5) = 0
(h² - 2) = 0

h² = -5
h² = 2

h = +/- √-5, √2

I say "+/- √-5" because either number could be used, as both a positive and negative number will be required to exponentiate into -5.

h^4 + 3h^2 - 10
= h^4 + 5h^2 - 2h^2 - 10
= (h^4 + 5h^2) - (2h^2 + 10)
= h^2(h^2 + 5) - 2(h^2 + 5)
= (h^2 + 5)(h^2 - 2)

( h ² + 5 ) ( h ² - 2 )

(h^2+5)(h^2-2)

(h^2 +5)(h^2 - 2)