Since there are no common factors for all three terms, we proceed with finding the value of two integers that multiply to ac = 2 \cdot 5 = 10 and add up to b = 11. After finding these integers, we use them to rewrite the middle term 11x as a sum of two terms, and then factor the trinomial by grouping.

The factors of 10 are 1, 2, 5, and 10. Among these factors, 10 and 1 are the pair that add up to 11.

We can use this to rewrite the trinomial and factor by grouping as follows:

There are no more factors to be taken out, so the fully factored form of the polynomial is \left(2x+1\right)\left(x+5\right).

We can check the answer by multiplying the factored form \left(2x+1\right)\left(x+5\right).

Also note that we could have also rewritten the polynomial as 2x^2 + 10x + x + 5. This would have resulted in a different middle step in factoring by grouping, but the same end result.

Since there are no common factors for all three terms, we proceed with finding the value of two integers that multiply to ac = 5 \cdot 9 = 45 and add up to b = -18. After finding these integers, we use them to rewrite the middle term -18x as a sum of two terms, and then factor the trinomial by grouping.

The factors of 45 are \pm 1, \pm 3, \pm 5, \pm 9, \pm 15, and \pm 45. Note that since the middle term of the trinomial is negative, we need to consider negative factors as well as positive ones. Of these factors, -15 and -3 are the pair that add up to -18.

We can use this to rewrite the trinomial and factor by grouping as follows:

There are no more factors to be taken out, so the fully factored form of the polynomial is \left(x - 3\right)\left(5x - 3\right).

We can check the answer by multiplying the factored form \left(5x-3\right)\left(x-3\right).

This polynomial has a common factor between all three terms. We can start by taking out this factor, then proceeding to look for integers that we can use to rewrite the polynomial and use factoring by grouping.

First, we find any common factors for all three terms and factor it out. Then, from the simpler trinomial, we find the values of r and s, that multiply to ac and add up to b. After finding r and s, we write the trinomial in the form ax^{2} + rx + sx + c and factor it by grouping.

The greatest common factor of the three terms is 3. Taking out this factor, we have15x^2 - 27x - 6 = 3\left(5 x^2 - 9x - 2\right)

From the trinomial 5 x^2 - 9x - 2, we want to find two numbers that have a product of ac = \left(5\right)\left(-2\right) = -10 and a sum of b = -9. The factors of -10 include \pm 1, \pm 2, \pm 5, and \pm 10. Of these factors, -10 and 1 are the pair that add up to -9.

We can use this to rewrite the expression and factor by grouping as follows:

There are no more factors to be taken out, so the fully factored form of the polynomial is 3\left(5x + 1\right)\left(x - 2\right).

We can check the answer by multiplying the factored form 3\left(5x+1\right)\left(x-2\right).