Factor the polynomial

X^3 + 7x^2 +15x + 9

I tried factoring by grouping. I got stuck at this part:
x^2 ( x + 7 )...

To factor the given polynomial x^3 + 7x^2 + 15x + 9, you can use several methods such as factoring by grouping, synthetic division, or the rational root theorem. In this case, factoring by grouping is a suitable method.

You correctly factored out the greatest common factor, which is x^2 from the first two terms, resulting in x^2(x + 7). Now, let's focus on factoring the remaining two terms: 15x + 9.

To factor 15x + 9, we can look for two numbers that multiply to give 15 (the coefficient of x) and add up to give 9 (the constant term). The numbers that meet such criteria here are 3 and 5.

So, we can rewrite 15x + 9 as 3x + 5x + 9. Now, let's factor by grouping:

x^2(x + 7) + 3x + 5x + 9

Next, group the terms in pairs:

(x^2 + 3x) + (5x + 9)

Now, look for the greatest common factor in each group. In the first group, you can factor out x, and in the second group, you can factor out 5:

x(x + 3) + 5(x + 3)

Observe that both groups have a common factor of (x + 3). You can now factor out (x + 3) from both terms:

(x + 3)(x + 5).

Therefore, the factored form of the polynomial x^3 + 7x^2 + 15x + 9 is (x + 3)(x + 5).