Factor 21x² + 44x + 20

Here we will show you how to factor the quadratic function 21x² + 44x + 20 using the box method. In other words, we will show you how to factor 21x squared plus 44x plus 20 (21x^2 + 44x + 20) using the box method. It is a 5-step process:

Step 1: The standard form of a quadratic equation is ax² + bx + c. We start by labeling the different parts of our equation 21x² + 44x + 20, like this:

a = 21
b = 44
c = 20

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

2	14x	20
3x	21x²	30x
	7x	10

We put 21x² (a) in the bottom left square and 20 (c) in the top right square, like this:

2	14x	20
3x	21x²	30x
	7x	10

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 21 times 20 (a × c), and add together to equal 44 (b).

More specifically, 21 times 20 is 420. Therefore, we need to find the two numbers that multiply to equal 420, and add to equal 44.

? × ? = 420
? + ? = 44

After looking at this problem, we can see that the two numbers that multiply together to equal 420, and add together to equal 44, are 14 and 30, as illustrated here:

14 × 30 = 420
14 + 30 = 44

Now, we can fill in the last two squares in our box with 14x and 30x. Place 14x in the upper left square, and place 30x in the lower right square.

2	14x	20
3x	21x²	30x
	7x	10

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 14x and 20. The greatest common factor of 14x and 20 is 2. Therefore, we write 2 to the left of the top row. You can see it here in the color green:

2	14x	20
3x	21x²	30x
	7x	10

Next, let’s look at the bottom row. We have the terms 21x² and 30x. The greatest common factor of 21x² and 30x is 3x. Therefore, we write 3x to the left of the bottom row. You can see it here in the color blue:

2	14x	20
3x	21x²	30x
	7x	10

To find the values below the table, we first divide 21x² by 3x (labeled in blue). This gives us 7x.

21x² ÷ 3x = 7x

You can see this value colored in orange below:

2	14x	20
3x	21x²	30x
	7x	10

Next, we divide 30x by 3x (labeled in blue). This gives us 10.

30x ÷ 3x = 10

You can see this value colored in purple below:

2	14x	20
3x	21x²	30x
	7x	10

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 21x² + 44x + 20. You simply put the values on the left in one set of parentheses, and the values below in another set of parentheses, to get the answer:

(3x + 2)(7x + 10)

That’s it! Now you know how to factor the equation 21x² + 44x + 20.

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