Factor 4x² + 35x + 75

Here we will show you how to factor the quadratic function 4x² + 35x + 75 using the box method. In other words, we will show you how to factor 4x squared plus 35x plus 75 (4x^2 + 35x + 75) 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 4x² + 35x + 75, like this:

a = 4
b = 35
c = 75

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

15	15x	75
4x	4x²	20x
	x	5

We put 4x² (a) in the bottom left square and 75 (c) in the top right square, like this:

15	15x	75
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 75 (a × c), and add together to equal 35 (b).

More specifically, 4 times 75 is 300. Therefore, we need to find the two numbers that multiply to equal 300, and add to equal 35.

? × ? = 300
? + ? = 35

After looking at this problem, we can see that the two numbers that multiply together to equal 300, and add together to equal 35, are 15 and 20, as illustrated here:

15 × 20 = 300
15 + 20 = 35

Now, we can fill in the last two squares in our box with 15x and 20x. Place 15x in the upper left square, and place 20x in the lower right square.

15	15x	75
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 15x and 75. The greatest common factor of 15x and 75 is 15. Therefore, we write 15 to the left of the top row. You can see it here in the color green:

15	15x	75
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:

15	15x	75
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:

15	15x	75
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:

15	15x	75
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 + 35x + 75. 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:

(4x + 15)(x + 5)

That’s it! Now you know how to factor the equation 4x² + 35x + 75.

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