Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A 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]