To determine if $(x+2)$ is a factor of $x^3 + 8x^2 - 2x + 3$, we can apply synthetic division or use the factor theorem.

Using the Factor Theorem:

The factor theorem states that $(x + a)$ is a factor of a polynomial $P(x)$ if and only if $P(-a) = 0$.

For $(x + 2)$ to be a factor, we substitute $x = -2$ into the polynomial $P(x) = x^3 + 8x^2 - 2x + 3$ and check if the result is 0.

$P(-2) = (-2)^3 + 8(-2)^2 - 2(-2) + 3$ $P(-2) = -8 + 8(4) + 4 + 3$ $P(-2) = -8 + 32 + 4 + 3 = 31$

Since $P(-2) \neq 0$, $(x+2)$ is not a factor of $x^3 + 8x^2 - 2x + 3$.

Thus, the statement is false.

Would you like more details on this process?

Here are some related questions to explore:

1. What is the Remainder Theorem and how is it applied in polynomial division?
2. How can synthetic division be used to divide polynomials?
3. What are the conditions for a polynomial to be divisible by a linear factor?
4. How can you find all factors of a cubic polynomial?
5. Can a polynomial have irrational or complex factors?

Tip: When verifying whether a polynomial has a specific factor, always check the result of substituting the root associated with the factor into the polynomial.