solve the parabola equation 
x-4/-2 - 4x+5/2x+3 

The equation you provided:

\[
\frac{x - 4}{-2} - \frac{4x + 5}{2x + 3}
\]

This is a rational expression involving fractions. Let’s break it down and solve for \( x \).

1. **Simplify the first term:**

\[
\frac{x - 4}{-2} = -\frac{x - 4}{2} = -\frac{x}{2} + 2
\]

Now the expression becomes:

\[
-\frac{x}{2} + 2 - \frac{4x + 5}{2x + 3}
\]

2. **To subtract the fractions**, we need a common denominator. The common denominator between 2 and \( 2x + 3 \) is \( 2(2x + 3) \).

3. **Rewrite each fraction with the common denominator:**

For \( -\frac{x}{2} + 2 \), the new denominator becomes \( 2(2x + 3) \):

\[
-\frac{x}{2} + 2 = \frac{-(x)(2x + 3)}{2(2x + 3)} + \frac{2(2x + 3)}{2(2x + 3)}
\]

Now, the expression becomes:

\[
\frac{-(x)(2x + 3) + 2(2x + 3)}{2(2x + 3)} - \frac{4x + 5}{2x + 3}
\]

4. **Combine the fractions:**

Now the equation is:

\[
\frac{-(x)(2x + 3) + 2(2x + 3) - (4x + 5)}{2(2x + 3)}
\]

5. **Simplify the numerator:**

Distribute terms inside the brackets:

\[
-(x)(2x + 3) = -2x^2 - 3x
\]
\[
2(2x + 3) = 4x + 6
\]
Now, the numerator becomes:

\[
-2x^2 - 3x + 4x + 6 - (4x + 5)
\]

Simplify the terms:

\[
-2x^2 - 3x + 4x - 4x + 6 - 5 = -2x^2 + x + 1
\]

6. **Final expression:**

Now, the expression is:

\[
\frac{-2x^2 + x + 1}{2(2x + 3)}
\]

This is the simplified form of the original expression. If you're looking for the roots or further steps, let me know!

Would you like further details or have any questions?

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Here are 5 related questions to explore:

1. How do we solve rational equations like this?
2. How can you factor the numerator of a rational expression?
3. What does it mean to find the roots of a rational expression?
4. How does the denominator impact the domain of the equation?
5. What are common techniques for simplifying algebraic fractions?

**Tip**: Always check the domain of your equation to ensure no division by zero.

solve the parabola equation x-4/-2 - 4x+5/2x+3

Solve the rational expression x-4/-2 - 4x+5/2x+3 with a detailed step-by-step breakdown. This problem involves simplifying rational expressions using common denominators and algebraic manipulation. Suitable for students in Grades 9-11.

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