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(3 x^2 + 1) (x^2 - 2)

Step-by-step explanation:

Factor the following:

3 x^4 - 5 x^2 - 2

Hint: | Express 3 x^4 - 5 x^2 - 2 as a quadratic in x^2.

3 x^4 - 5 x^2 - 2 = 3 (x^2)^2 - 5 x^2 - 2:

3 (x^2)^2 - 5 x^2 - 2

Hint: | Factor 3 (x^2)^2 - 5 x^2 - 2 which is quadratic with respect to x^2.

Factor the quadratic 3 (x^2)^2 - 5 x^2 - 2. The coefficient of x^4 is 3 and the constant term is -2. The product of 3 and -2 is -6. The factors of -6 which sum to -5 are 1 and -6. So 3 (x^2)^2 - 5 x^2 - 2 = 3 x^4 - 6 x^2 + x^2 - 2 = x^2 (3 x^2 + 1) - 2 (3 x^2 + 1):

x^2 (3 x^2 + 1) - 2 (3 x^2 + 1)

Hint: | Factor common terms from x^2 (3 x^2 + 1) - 2 (3 x^2 + 1).

Factor 3 x^2 + 1 from x^2 (3 x^2 + 1) - 2 (3 x^2 + 1):

(3 x^2 + 1) (x^2 - 2)