Let's simplify the expression:

\[
(4x^2 - 6x + 1) - (5x^2 + 8x + 6)
\]

Distribute the negative sign to each term inside the second parenthesis:

\[
4x^2 - 6x + 1 - 5x^2 - 8x - 6
\]

Now, combine like terms:

- For the \(x^2\) terms: \(4x^2 - 5x^2 = -x^2\)
- For the \(x\) terms: \(-6x - 8x = -14x\)
- For the constant terms: \(1 - 6 = -5\)

So, the simplified expression is:

\[
-x^2 - 14x - 5
\]

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

1. How would you solve the equation \( -x^2 - 14x - 5 = 0 \)?
2. What are the roots of the quadratic equation \( x^2 + 14x + 5 = 0 \)?
3. How do you complete the square for \( -x^2 - 14x - 5 \)?
4. What is the vertex form of the quadratic equation \( -x^2 - 14x - 5 \)?
5. How can you graph the quadratic function \( -x^2 - 14x - 5 \)?

**Tip:** When combining like terms, be careful with signs, especially when distributing a negative sign across a polynomial.

Learn how to simplify the expression (4x^2 - 6x + 1) - (5x^2 + 8x + 6) step-by-step with a detailed solution. This problem involves algebraic simplification of polynomials, focusing on combining like terms and distributing negative signs. Suitable for students in Grades 9-12.

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