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when bringing x to one side from '-5x-2=x^2+2' i did: -x^2-5x-4=0. I then factorised this and got
-(x-4)(x-1)=0. Thus, when finding the final two values i ended up ignoring the rule of inverting the operation sign in the brackets because there was already a negative sign on the outside of the bracket. Can I be correct in doing this?

Let's analyze your work step by step:

Given:
\[ f(-x) = h(x) + 2 \]

The provided steps are:

1. \[ 5(-x) - 2 = x^2 + 2 \]
2. \[ -5x - 2 = x^2 + 2 \]
3. \[ x^2 + 5x + 4 = 0 \]

So far, the work is correct.

Next:
\[ (x + 4)(x + 1) = 0 \]

This step correctly factorizes the quadratic equation \( x^2 + 5x + 4 = 0 \).

Finally:
\[ x = -4 \quad \text{or} \quad x = -1 \]

This is the correct solution set for the quadratic equation.

However, your method and notation seem to be a bit different. Let's clarify your approach:

From the step:
\[ -5x - 2 = x^2 + 2 \]

Moving all terms to one side gives:
\[ x^2 + 5x + 4 = 0 \]

To factorize:
\[ (x + 4)(x + 1) = 0 \]

So,
\[ x = -4 \quad \text{or} \quad x = -1 \]

If your alternative method was:
\[ -x^2 - 5x - 4 = 0 \]

Factorizing would be:
\[ - (x^2 + 5x + 4) = 0 \]
\[ - (x + 4)(x + 1) = 0 \]

By multiplying both sides by -1, you get:
\[ (x + 4)(x + 1) = 0 \]

Thus:
\[ x = -4 \quad \text{or} \quad x = -1 \]

Both methods arrive at the same solution. Therefore, your intuition about the signs and brackets was correct. You can apply the rule of inverting the signs inside the brackets as needed, and it will not change the solution in this context.

Would you like more details or have any other questions?

Here are some related questions to further explore the topic:

1. How do you derive the quadratic formula from a general quadratic equation?
2. What are the properties of polynomial functions of different degrees?
3. How do you factorize cubic equations?
4. What are the different methods to solve quadratic equations?
5. How do transformations affect the graph of a polynomial function?
6. What are the different types of roots a polynomial equation can have?
7. How can you determine if a quadratic equation will have real or complex roots?
8. What is the importance of the discriminant in solving quadratic equations?

**Tip:** When solving quadratic equations, always check if the quadratic can be easily factorized before using the quadratic formula or completing the square. This can save time and simplify the process.

Explore how to solve the quadratic equation -5x - 2 = x^2 + 2 step-by-step. Learn about factoring quadratic equations and the correct application of sign inversion rules. Suitable for high school students and those studying advanced algebra.

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