Let's solve this step-by-step.

1. Identify the formula for the volume of a rectangular prism: $\text{Volume} = \text{length} \times \text{width} \times \text{height}$

2. Substitute the given values:

   • Length = $(x + 2)$ cm
   • Width = $x$ cm
   • Height = $1$ cm
   • Volume = $15$ cm³

   So, we can set up the equation: $(x + 2) \times x \times 1 = 15$

3. Simplify the equation: $(x + 2) x = 15$ Expanding it gives: $x^2 + 2x = 15$

4. Rearrange the equation to standard quadratic form: $x^2 + 2x - 15 = 0$

5. Factor the quadratic equation: We need to find two numbers that multiply to -15 and add up to 2. The factors are 5 and -3.

   So, we factor the equation as: $(x + 5)(x - 3) = 0$

6. Solve for $x$: Set each factor equal to zero:

   • $x + 5 = 0 \Rightarrow x = -5$
   • $x - 3 = 0 \Rightarrow x = 3$

   Since width (x) cannot be negative, we discard $x = -5$ and take $x = 3$.

7. Conclusion: The width of the box is $3$ cm.

Filling in the blanks:

• The factored form is $(x + 5)(x - 3) = 0$.
• Width $= 3$ cm.

Would you like further details on each step or have any questions?

Here are some related questions to expand on this problem:

1. How would the equation change if the height were 2 cm instead of 1 cm?
2. Can you solve for length if only width and height are known?
3. What would happen to the width if the volume were increased to 30 cm³?
4. How would you factor an equation if it couldn't be easily factored by inspection?
5. How does this approach apply to other 3D shapes, like cylinders or cones?

Tip: Always check the feasibility of the solutions, especially in real-world problems, to ensure they make sense physically.