Fully factorise the expression (x+2)^3-x^3

To fully factorize the expression \((x+2)^3 - x^3\), we can use the identity for the difference of cubes, which states:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

Here, let \(a = x+2\) and \(b = x\). Applying the identity:

1. **Identify \(a\) and \(b\):**
   \[
   a = x + 2 \quad \text{and} \quad b = x
   \]

2. **Apply the difference of cubes formula:**
   \[
   (x+2)^3 - x^3 = (a^3 - b^3) = (a - b)(a^2 + ab + b^2)
   \]

   Substitute \(a\) and \(b\) into the formula:
   \[
   (x+2)^3 - x^3 = ((x+2) - x)((x+2)^2 + (x+2)x + x^2)
   \]

3. **Simplify the expression:**
   \[
   (x+2)^3 - x^3 = (2)((x+2)^2 + (x+2)x + x^2)
   \]

4. **Expand and simplify the polynomial inside the parentheses:**
   \[
   (x+2)^2 = x^2 + 4x + 4
   \]
   \[
   (x+2)x = x^2 + 2x
   \]
   \[
   \text{So,} \quad (x+2)^2 + (x+2)x + x^2 = x^2 + 4x + 4 + x^2 + 2x + x^2
   \]
   Combine like terms:
   \[
   x^2 + 4x + 4 + x^2 + 2x + x^2 = 3x^2 + 6x + 4
   \]

5. **Substitute back:**
   \[
   (x+2)^3 - x^3 = 2(3x^2 + 6x + 4)
   \]

Therefore, the fully factorized form of \((x+2)^3 - x^3\) is:
\[
(x+2)^3 - x^3 = 2(3x^2 + 6x + 4)
\]

Would you like more details or have any questions?

Here are some questions you might find useful:
1. Can you explain the difference of cubes formula in more detail?
2. How do you simplify polynomial expressions?
3. What are other common algebraic identities used in factorization?
4. Can you show more examples of factorizing expressions using identities?
5. How do you check if a factorization is correct?

**Tip:** Understanding and recognizing algebraic identities, like the difference of cubes, can simplify complex polynomial expressions and make factorization easier.

Learn how to fully factorise the expression (x+2)^3 - x^3 using the difference of cubes formula. This problem covers algebraic concepts such as polynomials and factorization, suitable for Grades 9-12 students.

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