At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.

Multiply the following:

[tex]\[ \frac{x^2}{x+5} \cdot \frac{x^2 + 4x - 5}{x^2 - 4x} \][/tex]

Sagot :

Sure, let's solve this step-by-step!

Given the problem:
[tex]\[ \frac{x^2}{x+5} \cdot \frac{x^2 + 4x - 5}{x^2 - 4x} \][/tex]

Step 1: Express the fractions separately.

First fraction:
[tex]\[ \frac{x^2}{x+5} \][/tex]

Second fraction:
[tex]\[ \frac{x^2 + 4x - 5}{x^2 - 4x} \][/tex]

Step 2: Multiply the fractions.

[tex]\[ \frac{x^2}{x+5} \cdot \frac{x^2 + 4x - 5}{x^2 - 4x} = \frac{x^2 (x^2 + 4x - 5)}{(x + 5)(x^2 - 4x)} \][/tex]

Step 3: Simplify the resulting expression, if possible.

To simplify, let's factorize the numerator and the denominator:
- The numerator \(x^2 (x^2 + 4x - 5)\) is already factored as much as possible.
- The denominator \( (x + 5)(x^2 - 4x) \) can be factored further.

Notice that \( x^2 - 4x \) can be written as \( x(x - 4) \).

So, the expression becomes:
[tex]\[ \frac{x^2 (x^2 + 4x - 5)}{(x + 5)x(x - 4)} \][/tex]

Next, let's factorize \(x^2 + 4x - 5\). This factors to \( (x + 5)(x - 1) \).

Thus the expression now looks like:
[tex]\[ \frac{x^2 (x + 5)(x - 1)}{(x + 5)x(x - 4)} \][/tex]

Step 4: Cancel common factors in the numerator and the denominator.

The \( (x + 5) \) and one \( x \) in the numerator and denominator cancel out:
[tex]\[ \frac{x(x - 1)}{x - 4} \][/tex]

So, the simplified form of the original product is:
[tex]\[ \frac{x(x - 1)}{x - 4} \][/tex]

Therefore, the final simplified result is:
[tex]\[ \frac{x(x - 1)}{x - 4} \][/tex]