To solve the problem:

We are tasked to find the product of $2x^2 + 6x - 8$ and $x + 3$ and express the result in standard form.

Steps:

1. Distribute $x$ across $2x^2 + 6x - 8$: $x \cdot (2x^2 + 6x - 8) = 2x^3 + 6x^2 - 8x$

2. Distribute $3$ across $2x^2 + 6x - 8$: $3 \cdot (2x^2 + 6x - 8) = 6x^2 + 18x - 24$

3. Add the results together: $(2x^3 + 6x^2 - 8x) + (6x^2 + 18x - 24)$

4. Combine like terms: $2x^3 + (6x^2 + 6x^2) + (-8x + 18x) + (-24)$ $2x^3 + 12x^2 + 10x - 24$

Final Answer:

$2x^3 + 12x^2 + 10x - 24$

Would you like a breakdown of any specific step, or have any further questions?

Here are 5 related questions to expand understanding:

1. How can you verify the result of polynomial multiplication?
2. What is the degree of the resulting polynomial, and why?
3. What does "standard form" mean in the context of polynomials?
4. How would the process change if the second polynomial were $x - 3$ instead of $x + 3$?
5. Can you factorize the resulting polynomial further? If yes, how?

Tip: Always combine like terms carefully to ensure accuracy when simplifying polynomial expressions!