1) Multiply and simplify the following: x+2x−1 ⋅ x−3x+2 (1 point) Responses x−1 x minus 1 1x+2 the fraction with numerator 1 and denominator x plus 2 x−3x−1 the fraction with numerator x minus 3 and denominator x minus 1 1x−1

To multiply and simplify the expression \( (x + 2x - 1) \cdot (x - 3x + 2) \), we first simplify each expression inside the parentheses:

1. Combine like terms in the first parentheses: \[ x + 2x - 1 = 3x - 1 \]

2. Combine like terms in the second parentheses: \[ x - 3x + 2 = -2x + 2 \]

Now we can rewrite the product: \[ (3x - 1)(-2x + 2) \]

Next, we apply the distributive property (FOIL method): \[ (3x)(-2x) + (3x)(2) + (-1)(-2x) + (-1)(2) \]

Calculating each term:

• \( 3x \cdot -2x = -6x^2 \)
• \( 3x \cdot 2 = 6x \)
• \( -1 \cdot -2x = 2x \)
• \( -1 \cdot 2 = -2 \)

Now combine the results: \[ -6x^2 + 6x + 2x - 2 \]

Combining the \( x \) terms: \[ -6x^2 + 8x - 2 \]

Thus, the simplified product is: \[ \boxed{-6x^2 + 8x - 2} \]

Answer? question is mutiply and simplify the follwoing x+2 over x-1 mutiplied by x-3 over x+2

To multiply and simplify the expression \(\frac{x + 2}{x - 1} \cdot \frac{x - 3}{x + 2}\), follow these steps:

1. Multiply the fractions: \[ \frac{x + 2}{x - 1} \cdot \frac{x - 3}{x + 2} = \frac{(x + 2)(x - 3)}{(x - 1)(x + 2)} \]

2. Notice that \(x + 2\) in the numerator and denominator cancels out, assuming \(x \neq -2\): \[ = \frac{(x - 3)}{(x - 1)} \]

Thus, the simplified expression is: \[ \frac{x - 3}{x - 1} \]

So the final answer is: \[ \boxed{\frac{x - 3}{x - 1}} \]

Just remember that \(x\) cannot equal \(-2\) (which would make the original fraction undefined) or \(1\) (which would make the denominator in the second fraction undefined).

Divide and simplify x^2-4 over x+1 divided by x-2 over x+1

To divide and simplify the expression \(\frac{x^2 - 4}{x + 1} \div \frac{x - 2}{x + 1}\), we can follow these steps:

1. Rewrite the division as multiplication: Division by a fraction is equivalent to multiplying by its reciprocal: \[ \frac{x^2 - 4}{x + 1} \div \frac{x - 2}{x + 1} = \frac{x^2 - 4}{x + 1} \cdot \frac{x + 1}{x - 2} \]

2. Notice that \(x + 1\) cancels out: As long as \(x \neq -1\): \[ = \frac{x^2 - 4}{x - 2} \]

   The expression \(x^2 - 4\) can be factored as a difference of squares: \[ x^2 - 4 = (x - 2)(x + 2) \]

3. Substitute the factored form: \[ = \frac{(x - 2)(x + 2)}{x - 2} \]

4. Cancel \(x - 2\): As long as \(x \neq 2\): \[ = x + 2 \]

So, the simplified expression is: \[ \boxed{x + 2} \]

Note that \(x\) cannot be \(2\) or \(-1\) to avoid undefined expressions in the original problem.

Add and simplify x+1 over x^2-1 plus x-1 over x^2-1

To add and simplify the expression \(\frac{x + 1}{x^2 - 1} + \frac{x - 1}{x^2 - 1}\), we can follow these steps:

1. Identify a common denominator:
   Since both fractions have the same denominator, we can add them directly: \[ \frac{x + 1 + (x - 1)}{x^2 - 1} \]

2. Combine the numerators: \[ = \frac{x + 1 + x - 1}{x^2 - 1} \]

   Simplifying the numerator: \[ = \frac{2x}{x^2 - 1} \]

3. Factor the denominator: The denominator \(x^2 - 1\) is a difference of squares, which can be factored as: \[ x^2 - 1 = (x - 1)(x + 1) \]

4. Rewrite the expression: So now we have: \[ = \frac{2x}{(x - 1)(x + 1)} \]

This expression is already simplified, as \(2x\) does not have any common factors with the expression in the denominator.

Final Answer

The final simplified expression is: \[ \boxed{\frac{2x}{(x - 1)(x + 1)}} \]

Note that \(x\) cannot equal \(1\) or \(-1\) in order to avoid division by zero in the original expression.

Um can you look back at your work I dont think its right because thats not an option opitions are 2x over x-1
2x over x+1
2x over x^2-1
2x+2 over x-1

Let’s re-evaluate the expression \(\frac{x + 1}{x^2 - 1} + \frac{x - 1}{x^2 - 1}\) to clarify the simplification:

1. Identify the common denominator:
   Since both fractions share the same denominator (\(x^2 - 1\)), we can combine them directly: \[ \frac{x + 1 + (x - 1)}{x^2 - 1} \]

2. Combine the numerators: \[ = \frac{(x + 1) + (x - 1)}{x^2 - 1} = \frac{x + 1 + x - 1}{x^2 - 1} \] \[ = \frac{2x}{x^2 - 1} \]

3. Factor the denominator: The denominator \(x^2 - 1\) can be factored using the difference of squares: \[ x^2 - 1 = (x - 1)(x + 1) \]

4. Combine the expression: Therefore, we have: \[ = \frac{2x}{(x - 1)(x + 1)} \]

Now, let's look at the provided options again:

• \(\frac{2x}{x - 1}\)
• \(\frac{2x}{x + 1}\)
• \(\frac{2x}{x^2 - 1}\)
• \(\frac{2x + 2}{x - 1}\)

The expression \(\frac{2x}{(x - 1)(x + 1)}\) can also be expressed as: \[ \frac{2x}{x^2 - 1} \]

This means the correct option is: \[ \frac{2x}{x^2 - 1} \]

Final Answer

Thus, the correct answer is indeed: \[ \boxed{\frac{2x}{x^2 - 1}} \]

Thank you for your patience and for allowing me to clarify my response!

Add and simplify 1 over x+2 plus 3 over x-2

To add and simplify the expression \(\frac{1}{x + 2} + \frac{3}{x - 2}\), follow these steps:

1. Find a common denominator:
   The common denominator for the two fractions is \((x + 2)(x - 2)\).

2. Rewrite each fraction with the common denominator: \[ \frac{1}{x + 2} = \frac{1 \cdot (x - 2)}{(x + 2)(x - 2)} = \frac{x - 2}{(x + 2)(x - 2)} \] \[ \frac{3}{x - 2} = \frac{3 \cdot (x + 2)}{(x - 2)(x + 2)} = \frac{3(x + 2)}{(x - 2)(x + 2)} = \frac{3x + 6}{(x - 2)(x + 2)} \]

3. Combine the fractions: Now that both fractions have the same denominator, we can add them: \[ \frac{x - 2 + (3x + 6)}{(x + 2)(x - 2)} = \frac{x - 2 + 3x + 6}{(x + 2)(x - 2)} \]

4. Combine like terms in the numerator: \[ = \frac{(x + 3x) + (-2 + 6)}{(x + 2)(x - 2)} = \frac{4x + 4}{(x + 2)(x - 2)} \]

5. Factor the numerator: Notice that the numerator can be factored: \[ = \frac{4(x + 1)}{(x + 2)(x - 2)} \]

This cannot be simplified further since \(4\) and the expression in the denominator do not share any common factors.

Final Answer

Thus, the final simplified expression is: \[ \boxed{\frac{4(x + 1)}{(x + 2)(x - 2)}} \]

answers are 4x+4 over x^2-4
x-2 over x+2
x+2 over 4x
4x over x+2

Let's go through the addition and simplification process again for the expression \( \frac{1}{x + 2} + \frac{3}{x - 2} \) to verify the previous work with the given answer options.

1. Find a common denominator:
   The common denominator for the two fractions is \( (x + 2)(x - 2) \).

2. Rewrite each fraction with the common denominator: \[ \frac{1}{x + 2} = \frac{1 \cdot (x - 2)}{(x + 2)(x - 2)} = \frac{x - 2}{(x + 2)(x - 2)} \] \[ \frac{3}{x - 2} = \frac{3 \cdot (x + 2)}{(x - 2)(x + 2)} = \frac{3(x + 2)}{(x - 2)(x + 2)} = \frac{3x + 6}{(x - 2)(x + 2)} \]

3. Combine the fractions: Now we can add the two fractions: \[ \frac{x - 2 + (3x + 6)}{(x + 2)(x - 2)} = \frac{x - 2 + 3x + 6}{(x + 2)(x - 2)} \]

4. Combine like terms in the numerator: \[ = \frac{(x + 3x) + (-2 + 6)}{(x + 2)(x - 2)} = \frac{4x + 4}{(x + 2)(x - 2)} \]

5. Recognize that the denominator \( (x + 2)(x - 2) \) can be expanded: \[ = \frac{4x + 4}{x^2 - 4} \] (since \( x^2 - 4 \) represents the product of \( (x + 2)(x - 2) \))

Now, the simplified result can indeed be expressed as: \[ \frac{4x + 4}{x^2 - 4} \]

Final Answer

Therefore, the correct option is: \[ \boxed{\frac{4x + 4}{x^2 - 4}} \]

Thank you for your patience, and I apologize for any confusion in the previous explanations.