To divide the polynomials (4x^3 + 3x^2 + 2x - 1) by (x^3 + x - 2), we perform polynomial long division:

First, we divide the highest degree terms:

4x^3 / x^3 = 4

Next, we multiply the divisor (x^3 + x - 2) by the quotient (4) and subtract it from the dividend (4x^3 + 3x^2 + 2x - 1):

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

(4x^3 + 3x^2 + 2x - 1) - (4x^3 + 4x - 8) = 3x^2 - 2x + 7

Next, we bring down the next term:

3x^2 / x^3 = 3x

Next, we multiply the divisor (x^3 + x - 2) by the new term (3x) and subtract it from the previous result:

(3x)(x^3 + x - 2) = 3x^4 + 3x^2 - 6x

(3x^2 - 2x + 7) - (3x^4 + 3x^2 - 6x) = -3x^4 - x^2 + 4x + 7

At this point, the degree of the remaining polynomial is lower than the divisor, so we cannot continue the division. Therefore, the result of the division is:

4x + 3, with a remainder of -3x^4 - x^2 + 4x + 7.