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]
