Factor -4x² - 28x

Factor -4x² - 28x - 40

Here we will show you how to factor the quadratic function -4x² - 28x - 40 using the box method. In other words, we will show you how to factor negative 4x squared minus 28x minus 40 (-4x^2 - 28x - 40) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. In this equation, a is negative. Therefore, we need to start by setting aside the negative (-). We do this by flipping the signs in the equation to get 4x² + 28x + 40. Now we can label the different parts of our equation, like this:

a = 4
b = 28
c = 40

Step 2: Next, we need to draw a box and divide it into four squares:

8	8x	40
4x	4x²	20x
	x	5

We put 4x² (a) in the bottom left square and 40 (c) in the top right square, like this:

8	8x	40
4x	4x²	20x
	x	5

Step 3: In order to fill in the other two squares, we need to do some math. We have to find two numbers that multiply together to give us the product of 4 times 40 (a × c), and add together to equal 28 (b).

More specifically, 4 times 40 is 160. Therefore, we need to find the two numbers that multiply to equal 160, and add to equal 28.

? × ? = 160
? + ? = 28

After looking at this problem, we can see that the two numbers that multiply together to equal 160, and add together to equal 28, are 8 and 20, as illustrated here:

8 × 20 = 160
8 + 20 = 28

Now, we can fill in the last two squares in our box with 8x and 20x. Place 8x in the upper left square, and place 20x in the lower right square.

8	8x	40
4x	4x²	20x
	x	5

Step 4: Next, we need to find four final numbers to finish factoring our equation. We can see that our box is a 2 x 2 table made up of rows and columns.

Our goal is to find the numbers that, when multiplied together, result in the products in each square. We can do this by finding the greatest common factor in each row.

Let’s look at the top row. We have the terms 8x and 40. The greatest common factor of 8x and 40 is 8. Therefore, we write 8 to the left of the top row. You can see it here in the color green:

8	8x	40
4x	4x²	20x
	x	5

Next, let’s look at the bottom row. We have the terms 4x² and 20x. The greatest common factor of 4x² and 20x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

8	8x	40
4x	4x²	20x
	x	5

To find the values below the table, we first divide 4x² by 4x (labeled in blue). This gives us x.

4x² ÷ 4x = x

You can see this value colored in orange below:

8	8x	40
4x	4x²	20x
	x	5

Next, we divide 20x by 4x (labeled in blue). This gives us 5.

20x ÷ 4x = 5

You can see this value colored in purple below:

8	8x	40
4x	4x²	20x
	x	5

Step 5: Now we have all of the information we need to finish factoring the equation. The values outside of the box are used to factor -4x² - 28x - 40. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get:

(4x + 8)(x + 5)

In our original quadratic equation, -4x² - 28x - 40, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(4x + 8)(x + 5)

That’s it! Now you know how to factor the equation -4x² - 28x - 40.

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