Solve. 4x^3+2x^2-3x+1/(2x+1)(x-3)

To simplify the expression (4x^3+2x^2-3x+1)/(2x+1)(x-3), we'll start by factoring the numerator and the denominator.

Step 1: Factor the numerator
The numerator of the expression is 4x^3+2x^2-3x+1. To factor it, we can look for any common factors among the terms. In this case, there is no common factor among all the terms. So, we move on to the next step.

Step 2: Factor the denominator
The denominator of the expression is (2x+1)(x-3). It is already factored, so we can move on to the next step.

Step 3: Simplify by canceling common factors
To simplify the expression, we can cancel out any common factors between the numerator and the denominator.

To do this, we check if any of the factors in the numerator appear in the denominator, and vice versa.

In this case, we observe that the factor (2x+1) appears in both the numerator and the denominator. So, we can cancel it out.

(4x^3+2x^2-3x+1)/(2x+1)(x-3)
= [(2x+1)*(2x^2-x+1)]/(2x+1)(x-3)

Now, we no longer have any common factors to cancel out. So, we have simplified the expression as much as possible.

Therefore, the simplified form of (4x^3+2x^2-3x+1)/(2x+1)(x-3) is (2x^2-x+1)/(x-3).