factor of x^3 + 4x^2 + x - 6

To find the factors of the polynomial x^3 + 4x^2 + x - 6, we can use synthetic division or guess and check.

Using guess and check, we start by looking for a factor that could possibly divide the polynomial evenly. We can try to substitute different values for x to see if any of them result in a zero remainder. Let's start with x = 1:

(1)^3 + 4(1)^2 + (1) - 6 = 1 + 4 + 1 - 6 = 0

Since x = 1 is a zero of the polynomial, we can conclude that (x - 1) is a factor.

Now we can divide the polynomial by (x - 1) using synthetic division:
```
1 | 1 4 1 -6
- 1 -3 -2
_______________
1 3 -2 -8
```
The result is 1x^2 + 3x - 2 with a remainder of -8.

We now have a quadratic polynomial 1x^2 + 3x - 2. To find the remaining factors, we can try to solve the quadratic equation using factoring, completing the square, or the quadratic formula.

This quadratic equation can be factored as:
(x - 1)(x + 2)(x + 4)

Therefore, the factors of the polynomial x^3 + 4x^2 + x - 6 are: (x - 1), (x + 2), and (x + 4).