Factor -4x² - 27x

Factor -4x² - 27x - 44

Here we will show you how to factor the quadratic function -4x² - 27x - 44 using the box method. In other words, we will show you how to factor negative 4x squared minus 27x minus 44 (-4x^2 - 27x - 44) 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 + 27x + 44. Now we can label the different parts of our equation, like this:

a = 4
b = 27
c = 44

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

11	11x	44
4x	4x²	16x
	x	4

We put 4x² (a) in the bottom left square and 44 (c) in the top right square, like this:

11	11x	44
4x	4x²	16x
	x	4

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 44 (a × c), and add together to equal 27 (b).

More specifically, 4 times 44 is 176. Therefore, we need to find the two numbers that multiply to equal 176, and add to equal 27.

? × ? = 176
? + ? = 27

After looking at this problem, we can see that the two numbers that multiply together to equal 176, and add together to equal 27, are 11 and 16, as illustrated here:

11 × 16 = 176
11 + 16 = 27

Now, we can fill in the last two squares in our box with 11x and 16x. Place 11x in the upper left square, and place 16x in the lower right square.

11	11x	44
4x	4x²	16x
	x	4

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 11x and 44. The greatest common factor of 11x and 44 is 11. Therefore, we write 11 to the left of the top row. You can see it here in the color green:

11	11x	44
4x	4x²	16x
	x	4

Next, let’s look at the bottom row. We have the terms 4x² and 16x. The greatest common factor of 4x² and 16x is 4x. Therefore, we write 4x to the left of the bottom row. You can see it here in the color blue:

11	11x	44
4x	4x²	16x
	x	4

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:

11	11x	44
4x	4x²	16x
	x	4

Next, we divide 16x by 4x (labeled in blue). This gives us 4.

16x ÷ 4x = 4

You can see this value colored in purple below:

11	11x	44
4x	4x²	16x
	x	4

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 - 27x - 44. 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 + 11)(x + 4)

In our original quadratic equation, -4x² - 27x - 44, a is negative. Therefore, we need to add a negative (-) sign before our two sets of parentheses, like this:

-(4x + 11)(x + 4)

That’s it! Now you know how to factor the equation -4x² - 27x - 44.

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