Final answer:
To factor the expression 4x³ + 10x² - 6x, first factor out the greatest common factor, which is 2x, to get 2x(2x² + 5x - 3). Further factorization yields the complete factored form as 2x(2x - 1)(x + 3).
Step-by-step explanation:
The completely factored form of 4x³ + 10x² - 6x can be found by first factoring out the greatest common factor (GCF), which in this case is 2x. Doing so, we get:
2x(2x² + 5x - 3)
To factored the quadratic expression 2x² + 5x - 3, we look for two numbers that multiply to give 2*-3 = -6 and add to give 5. These numbers are 6 and -1. We rewrite the middle term 5x using 6x and -1x and factor by grouping:
2x[(2x² + 6x) + (-1x - 3)]
2x[2x(x + 3) -1(x + 3)]
2x(2x - 1)(x + 3)
Therefore, the factored form of the given polynomial 4x³ + 10x² - 6x is 2x(2x - 1)(x + 3).