Let's first simplify both expressions to determine whether they are equivalent or not.

Expression 1:

$5x^2 - 2x - 4 + 6x + 3$ We can combine like terms: $5x^2 + (-2x + 6x) + (-4 + 3) = 5x^2 + 4x - 1$

Expression 2:

$6x^2 - 6x + 6 - x^2 + 10x - 7$ We can combine like terms: $(6x^2 - x^2) + (-6x + 10x) + (6 - 7) = 5x^2 + 4x - 1$

Both simplified expressions are: $5x^2 + 4x - 1$

Since both expressions simplify to the same form, the two expressions are equivalent.

Now, let's evaluate the answer choices:

• First option: "The two expressions are not equivalent because when $x = 2$, the two expressions do not have the same value." — This is incorrect, as the expressions are equivalent and will always have the same value for any $x$.

• Second option: "The two expressions are not equivalent because when they are simplified, they do not have the same coefficients for the $x^2$ and $x$ terms." — This is also incorrect because both expressions have the same coefficients after simplification.

• Third option: "They are equivalent because the sum of the constants is the same in both expressions." — This is correct, but it's not the most accurate description, as the equivalence is due to the entire expressions being the same, not just the constants.

• Fourth option: "They are equivalent because when $x = 2$, the two expressions have the same value." — This is true, but evaluating at $x = 2$ is not necessary to prove equivalence. They are equivalent for all $x$, not just at $x = 2$.

Best answer:

The third option is the most accurate in terms of explaining why the expressions are equivalent.

Would you like further details or have any questions?

Here are five related questions to explore this further:

1. How do you check if two expressions are equivalent for all values of $x$?
2. What is the process of simplifying algebraic expressions?
3. Can two expressions be equivalent even if they look different initially?
4. How can you verify algebraic equivalence using specific values of $x$?
5. Why is it important to simplify expressions when solving problems?

Tip: Always simplify expressions fully before comparing them, as it often reveals hidden equivalencies.