Let's solve and simplify the given expression step-by-step:
We need to simplify the expression:
[tex]\[ \frac{2x^2 - 10}{x + 1} \cdot \frac{x - 4}{4x^2 - 20} \][/tex]
### Step 1: Factorize where possible
First, we look at the components of the expression and see if we can factorize the numerators and denominators.
#### Numerator and denominator of the first fraction:
[tex]\[ 2x^2 - 10 \][/tex]
This can be factored as:
[tex]\[ 2(x^2 - 5) \][/tex]
Hence,
[tex]\[ \frac{2x^2 - 10}{x + 1} = \frac{2(x^2 - 5)}{x + 1} \][/tex]
#### Numerator and denominator of the second fraction:
[tex]\[ 4x^2 - 20 \][/tex]
This can be factored as:
[tex]\[ 4(x^2 - 5) \][/tex]
Hence,
[tex]\[ \frac{x - 4}{4x^2 - 20} = \frac{x - 4}{4(x^2 - 5)} \][/tex]
### Step 2: Combine the fractions
Now, substitute these factored forms back into the original expression:
[tex]\[ \left(\frac{2(x^2 - 5)}{x + 1}\right) \cdot \left(\frac{x - 4}{4(x^2 - 5)}\right) \][/tex]
### Step 3: Simplify
Notice that [tex]\((x^2 - 5)\)[/tex] appears in both the numerator and the denominator, so we can cancel it out:
[tex]\[ \frac{2 \cancel{(x^2 - 5)}}{x + 1} \cdot \frac{x - 4}{4 \cancel{(x^2 - 5)}} \][/tex]
This simplifies to:
[tex]\[ \frac{2}{x + 1} \cdot \frac{x - 4}{4} \][/tex]
Multiply the remaining parts:
[tex]\[ \frac{2(x - 4)}{4(x + 1)} \][/tex]
### Step 4: Simplify further
We can divide both the numerator and the denominator by 2:
[tex]\[ \frac{(x - 4)}{2(x + 1)} \][/tex]
Thus, the expression simplifies to:
[tex]\[ \frac{x - 4}{2(x + 1)} \][/tex]
So the simplified fraction is [tex]\(\frac{x - 4}{2(x + 1)}\)[/tex].
