factorise the polynomial x3 + 4x2 + x − 6

To factorise the polynomial x^3 + 4x^2 + x - 6, we first look for any common factors, but in this case, there are none.

Next, we try to factor by grouping. To do this, we need to split the middle term. We look for two numbers that multiply to (a*d) = -6 (the product of the coefficient of x^3 and the constant term) and add up to b = 4 (the coefficient of x^2).

The numbers that satisfy these conditions are 3 and 2, because 3*2 = 6 and 3 + 2 = 5.

So, we rewrite the polynomial as follows:

x^3 + 3x^2 + 2x^2 + x - 6

Now, we group the terms and factor by grouping:

x^2(x + 3) + 1(2x^2 + x - 6)

Now, we factor the second group:

x^2(x + 3) + 1(2x - 3)(x + 2)

So, the factorised form of the polynomial x^3 + 4x^2 + x - 6 is:

(x^2 + 1)(x + 3)(2x - 3)