Question

How do you factor `6x^4+7x^2-10`?

Answer 1

Put `x^2 = y`. The expression becomes;

`f(y) = 6y^2 + 7y - 10 = (y - p)(y - q)`.
Factor this by the new AC Method.

Converted expression: `y^2 + 7y - 60 = (y - p')(y - q')`. Compose factor pairs of -60. Proceed: ...(-4, 15)(-5, 12). This last sum is 12 - 5 = 7 = b. Then, p' = -5 and q' = 12. Consequently, p = p'/a = -5/6, and q = q'/a = 12/6 = 2.

`f(y) = (y - 5/6)(y + 2) = (6y - 5)(y + 2). Replace y by x^2.`

`f(x) = (6x^2 - 5)(x^2 + 2)`

Check:
Develop: 6x^4 + 12x^2 - 5x^2 - 10 = 6x^4 + 7x^2 - 10. Correct