Factor 21x² + 31x + 10

Here we will show you how to factor the quadratic function 21x² + 31x + 10 using the box method. In other words, we will show you how to factor 21x squared plus 31x plus 10 (21x^2 + 31x + 10) 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 + 31x + 10, like this:

a = 21
b = 31
c = 10

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

10	10x	10
21x	21x²	21x
	x	1

We put 21x² (a) in the bottom left square and 10 (c) in the top right square, like this:

10	10x	10
21x	21x²	21x
	x	1

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 10 (a × c), and add together to equal 31 (b).

More specifically, 21 times 10 is 210. Therefore, we need to find the two numbers that multiply to equal 210, and add to equal 31.

? × ? = 210
? + ? = 31

After looking at this problem, we can see that the two numbers that multiply together to equal 210, and add together to equal 31, are 10 and 21, as illustrated here:

10 × 21 = 210
10 + 21 = 31

Now, we can fill in the last two squares in our box with 10x and 21x. Place 10x in the upper left square, and place 21x in the lower right square.

10	10x	10
21x	21x²	21x
	x	1

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 10x and 10. The greatest common factor of 10x and 10 is 10. Therefore, we write 10 to the left of the top row. You can see it here in the color green:

10	10x	10
21x	21x²	21x
	x	1

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

10	10x	10
21x	21x²	21x
	x	1

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

21x² ÷ 21x = x

You can see this value colored in orange below:

10	10x	10
21x	21x²	21x
	x	1

Next, we divide 21x by 21x (labeled in blue). This gives us 1.

21x ÷ 21x = 1

You can see this value colored in purple below:

10	10x	10
21x	21x²	21x
	x	1

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 + 31x + 10. 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:

(21x + 10)(x + 1)

That’s it! Now you know how to factor the equation 21x² + 31x + 10.

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