To find the derivative of $(x^2 + 2)(3x - 2)$ with respect to $x$, we apply the product rule. The product rule states that if you have two functions $u(x)$ and $v(x)$, then:

$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

Here, let:

• $u(x) = x^2 + 2$
• $v(x) = 3x - 2$

Now, we find their derivatives:

• $u'(x) = 2x$ (derivative of $x^2 + 2$)
• $v'(x) = 3$ (derivative of $3x - 2$)

Now, applying the product rule:

$\frac{d}{dx}\left[(x^2 + 2)(3x - 2)\right] = u'(x)v(x) + u(x)v'(x)$

$= (2x)(3x - 2) + (x^2 + 2)(3)$

Now, expand each term:

$= 2x(3x - 2) + 3(x^2 + 2)$

$= 6x^2 - 4x + 3x^2 + 6$

Combine like terms:

$= 9x^2 - 4x + 6$

Thus, the derivative is:

$\boxed{9x^2 - 4x + 6}$

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What is the product rule, and how is it applied in differentiation?
2. How do you simplify expressions after applying the product rule?
3. How do you handle higher-order derivatives using the product rule?
4. How do you find the derivative of a quotient of two functions (quotient rule)?
5. What happens if you differentiate $(x^2 + 2)^2$ using the chain rule instead?

Tip: Always check your expansions and combine like terms carefully when differentiating polynomial products.