`(x^(3)+8)/(x+2)=x^(2)+2x+4`. Is the given statement true?

Answer

To determine if the statement \((x^3 + 8)/(x + 2) = x^2 + 2x + 4\) is true, we can simplify the left-hand side and see if it equals the right-hand side.

### Step-by-Step Solution:

1. **Identify the expression on the left-hand side:**
\[ \frac{x^3 + 8}{x + 2} \]

2. **Recognize that \(x^3 + 8\) can be factored using the sum of cubes formula:**
The sum of cubes formula states that:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Here, \(a = x\) and \(b = 2\). Therefore:
\[ x^3 + 8 = x^3 + 2^3 = (x + 2)(x^2 - 2x + 4) \]

3. **Substitute the factored form back into the expression:**
\[ \frac{(x + 2)(x^2 - 2x + 4)}{x + 2} \]

4. **Cancel out the common factor \((x + 2)\):**
Assuming \(x + 2 \neq 0\) (i.e., \(x \neq -2\)):
\[ = x^2 - 2x + 4 \]

5. **Compare the simplified left-hand side with the right-hand side:**
The right-hand side is:
\[ x^2 + 2x + 4 \]
Now we have:
\[ x^2 - 2x + 4 \quad \text{(from the left-hand side)} \]
and
\[ x^2 + 2x + 4 \quad \text{(from the right-hand side)} \]

6. **Check if the two expressions are equal:**
The left-hand side \(x^2 - 2x + 4\) is not equal to the right-hand side \(x^2 + 2x + 4\) because the coefficients of \(x\) are different.

### Conclusion:
The statement \(\frac{x^3 + 8}{x + 2} = x^2 + 2x + 4\) is **false**.