The given equation is:

\[
x^3 + 3x + 2 = 0
\]

To solve this cubic equation, let's try factoring or using trial and error to find a rational root.

### Step 1: Test for rational roots
We can use the Rational Root Theorem, which states that any rational root of the equation is a factor of the constant term (2) divided by a factor of the leading coefficient (1). So the possible rational roots are \(\pm 1, \pm 2\).

Let's test these values by substituting them into the equation.

- For \(x = -1\):

\[
(-1)^3 + 3(-1) + 2 = -1 - 3 + 2 = -2
\]
This is not a root.

- For \(x = -2\):

\[
(-2)^3 + 3(-2) + 2 = -8 - 6 + 2 = -12 + 2 = -10
\]
This is also not a root.

- For \(x = 1\):

\[
1^3 + 3(1) + 2 = 1 + 3 + 2 = 6
\]
This is not a root.

- For \(x = -1\):

\[
(-1)^3 + 3(-1) +2= ?......works= passes

This guide walks you through solving the cubic equation x^3 + 3x + 2 step-by-step using the Rational Root Theorem and factoring methods. Suitable for high school students, this explanation covers key algebraic concepts, including cubic equations and the Rational Root Theorem.

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