Factor into polynomials with integral coefficients: 6x^2+13x+6

integer or integral coefficents?

(3x+2)(2x+3)

Will (3x+2)(2x+3) work?

To factor the given polynomial, we need to find two binomials with integral coefficients whose product is equal to the original polynomial.

First, we look for two numbers whose product is equal to the product of the leading coefficient (6) and the constant term (6). In this case, the product is 6 * 6 = 36. We need to find a pair of numbers whose product is 36 and whose sum is equal to the coefficient of the middle term (13).

There are a few possible combinations that satisfy these conditions: (1, 36), (2, 18), (3, 12), and (4, 9).

Let's go through each possibility:

1. (1, 36): This means we would have (6x + 1)(x + 36). However, if we multiply this out, we do not get the original polynomial.

2. (2, 18): This means we would have (6x + 2)(x + 18). Again, if we multiply this out, we do not get the original polynomial.

3. (3, 12): This means we would have (6x + 3)(x + 12). Once again, if we multiply this out, we do not get the original polynomial.

4. (4, 9): This means we would have (6x + 4)(x + 9). If we multiply this out, we get 6x^2 + 49x + 36, which is not equal to the original polynomial.

Since none of the above combinations work, it means that the given polynomial does not factor into polynomials with integral coefficients.