In order to factor, we need to find two numbers that add up to , and multiply together to get .  Because the signs of those numbers are both negative, one of the numbers we're looking for will be positive, and one will be negative. It helps if we first look at all of the pairs of the factors of :

We know that these pairs will multiply to get , we just need to narrow it down to the pair that, when added, will give .  We find that the  pair does that.  We plug those numbers into our answer:

The first thing we can do is factor a  out of each term:

Now we need to find two numbers that add up to , and when multiplied make .  We can look at the pairs of factors of :

Of these pairs, only one will equal  when added together, so that's the one we're looking for.  To finish, we plug those numbers into our function:

Using the AC method, A=8,B=-2 C=-6, therefore A*C=-48. So you must find 2 factors of -48 which multiply and add/subtract to get the B term which is -2. So going through the factors of -48 you come to the factors of -8 and +6, which multiply together to give you -48 and add together to give you -2.

Then you insert factors as follows:

Factor by grouping which is as follows:

Then factor

Then

To solve this, first find what numbers give a product of 20 and when added yield -12. 

 Since the latter is negative, both numbers are negative. 

    and   

Therefore 

is the solution and can be verified by distributing:

To simplify the factorization pull out an x term

Find two numbers that add to 16 and multiply to 60.

To begin, let's clear up the  from the quadratic term by multiplying the entire numerator and denominator by :

We can distribute the  through the numerator:

Let's factor the  out of the denominator now, just to clear the problem up a bit:

Now that we have the equation in a more recognizable form, we need to find two numbers that add up to , and multiply to get .  Let's look at the factor pairs of :

We know that these pairs multiply to get  (if we give one of the numbers a negative sign), so we need to find the pair that adds to .  Because the three is negative, we know that the larger number will have the negative sign.  We find that:

So our numbers are  and .  Now we can plug them back into our equation to get a final answer of:

The polynomial will be easier to factor with an x pulled out

Two numbers are needed that add to 15 and multiply to 56. Trial and error is show that those two numbers are 7 and 8.

To factor this polynomial, first take a common factor of negative one in order to change the coefficient of  to positive.

Factor the terms inside the parentheses.

The possible factors of the last term, 42, that will provide the middle term   are .

Factorize the polynomial.

Distribute the negative one through the first binomial.

The answer is:  

The easiest way to solve this problem is to factor the original function, and then to find the zeroes from the factored form. To do this, we start with the original function, f(x). 

Next, we need to set up the function in factored form, leaving blanks for the numbers we don't yet know.

At this point, we need to find two numbers - one for each blank. By looking at the original function, we can gather a few clues that will help us find the two numbers. The product of these two numbers will be equal to the last term of our original function (-14, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of our original function (-5, or b in the standard quadratic formula). Because their product is negative (-14), that must mean that they have different signs - otherwise, their product would be positive. Also, because their sum is negative (-5), we know that the larger of the two numbers must be negative, otherwise their sum would be positive. 

Now, at this point, we may need to test a few different possibilities, using the clues we gathered from the original function. In the end, we'll find that the only numbers that work here are 2 and -7, as the product of 2 and -7 is -14, and sum of 2 and -7 is -5. So, this results in our function's factored form looking like...

Now, the final step in this problem is finding the zeros. To do this, we need to think about what a zero is. A zero is the x-value(s) at which...

So, to solve for our zeros, we just need to set the right side of our function equal to zero and solve for x. 

Because if either of these two factors is equal to zero, the entire function will be equal to zero (as anything multiplied by zero is zero), we can consider each of the two factors separately and solve for x. We'll start with the factor on the left.

We'll finish with the factor from the right.

Now, we have both of our zeros and the answer to our problem...

The easiest way to solve this problem is to factor the original function, and then to find the zeroes from the factored form. To do this, we start with the original function, f(x). 

Next, we need to set up the function in factored form, leaving blanks for the numbers we don't yet know.

At this point, we need to find two numbers - one for each blank. By looking at the original function, we can gather a few clues that will help us find the two numbers. The product of these two numbers will be equal to the last term of our original function (24, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of our original function (11, or b in the standard quadratic formula). Because their product is positive (24), that must mean that they have the same signs - otherwise, their product would be negative. Also, because their sum is positive (11), and we already know their signs are the same because their product is positive, we know that their signs must both be positive, as the sum of two negative numbers is negative.

Now, at this point, we may need to test a few different possibilities using the clues we gathered from the original function to guide our testing. In the end, we'll find that the only numbers that work here are 3 and 8, as the product of 3 and 8 is 24, and sum of 3 and 8 is 11. So, this results in our function's factored form looking like...

Now, the final step in this problem is finding the zeros. To do this, we need to think about what a zero is. A zero is the x-value(s) at which...

So, to solve for our zeros, we just need to set the right side of our function equal to zero and solve for x. 

Because if either of these two factors is equal to zero, the entire function will be equal to zero (as anything multiplied by zero is zero), we can consider each of the two factors separately and solve for x. We'll start with the factor on the left.

We'll finish with the factor from the right.

Now, we have both of our zeros and the answer to our problem...