Factor.

2x⁴ + x³ + 2x² + 4x - 24

tried x = ±1, ±2 and found x = -2 as a root

by long division ...
2x^4 + x^3 + 2x^2 + 4x - 24 = (x+2)(x^3 - 3x^2 + 8x - 12)

then tried ±1, ±2, ±3, ±4, ±1/2, ±3/2

quite tedious, but found x = 3/2
then another division....

2x^4 + x^3 + 2x^2 + 4x - 24 = (x+2)(3x-2)(2x^2 + 8)
= 2(x+2)(3x-2)(x^2 + 4)

To factor the expression 2x⁴ + x³ + 2x² + 4x - 24, we can use the method of factoring by grouping.

Step 1: Group the terms in pairs.
(2x⁴ + x³) + (2x² + 4x - 24)

Step 2: Factor out the greatest common factor (GCF) from each group.
x³(2x + 1) + 2x²(2x + 1) + 4x - 24

Step 3: Notice that we now have a common factor of (2x + 1). Factor this out.
(2x + 1)(x³ + 2x² + 4) - 24

Thus, the factored form of the expression 2x⁴ + x³ + 2x² + 4x - 24 is:
(2x + 1)(x³ + 2x² + 4) - 24