Find the possible length and width of a rectangular box with a volume of ( 6x^2 + 42x + 72 ) cubic centimeters if its height is 6 cm. Use the quadratic trinomial form which is x^2 + bx + c

We are given the volume of a rectangular box as \( V = 6x^2 + 42x + 72 \) cubic centimeters, and the height of the box as 6 cm. We need to find the possible length and width of the box.

### Step-by-Step Solution:

The volume \( V \) of a rectangular box is given by:
\[
V = \text{length} \times \text{width} \times \text{height}
\]

Let the length and width of the box be \( l \) and \( w \), respectively. Since the height is given as 6 cm, we can express the volume as:
\[
V = l \times w \times 6
\]

We are given that:
\[
6x^2 + 42x + 72 = l \times w \times 6
\]

To isolate \( l \times w \), divide both sides of the equation by 6:
\[
x^2 + 7x + 12 = l \times w
\]

Now, we need to factor the quadratic expression \( x^2 + 7x + 12 \):
\[
x^2 + 7x + 12 = (x + 3)(x + 4)
\]

Thus, \( l \times w = (x + 3)(x + 4) \). This means the possible values for the length and width are \( (x + 3) \) and \( (x + 4) \).

### Final Answer:
The possible length and width of the rectangular box are \( (x + 3) \) cm and \( (x + 4) \) cm.

Would you like more details or have any questions?

Here are five related questions:
1. How would the answer change if the height were different?
2. Can you apply this method to a different quadratic trinomial?
3. How do you factor more complex quadratic expressions?
4. What if the volume was given in terms of a different variable?
5. How does factoring help in solving real-world geometry problems?

**Tip:** When factoring quadratic expressions, always check if you can simplify the expression by dividing out common factors before proceeding.

Find the possible length and width of a rectangular box with a volume of (6x^2 + 42x + 72) cubic centimeters if its height is 6 cm. Use the quadratic trinomial form which is x^2 + bx + c.

Learn how to find the possible length and width of a rectangular box with a volume of 6x^2 + 42x + 72 cubic centimeters and a height of 6 cm. This problem focuses on factoring quadratic trinomials and applying algebraic concepts. Suitable for students in Grades 8-10.

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