Sure, let's solve the given expression step by step:
### Given Expression
The initial expression is:
[tex]\[
\frac{x^2 - 4x - 5}{x^2 - 3x + 2} \times \frac{x^2 - 5x + 6}{x^2 - 3x - 10}
\][/tex]
### Step 1: Factorize Each Polynomial
Let's factorize each of the polynomials as follows:
1. Numerator 1: [tex]\(x^2 - 4x - 5\)[/tex]
[tex]\[
x^2 - 4x - 5 = (x - 5)(x + 1)
\][/tex]
2. Denominator 1: [tex]\(x^2 - 3x + 2\)[/tex]
[tex]\[
x^2 - 3x + 2 = (x - 2)(x - 1)
\][/tex]
3. Numerator 2: [tex]\(x^2 - 5x + 6\)[/tex]
[tex]\[
x^2 - 5x + 6 = (x - 3)(x - 2)
\][/tex]
4. Denominator 2: [tex]\(x^2 - 3x - 10\)[/tex]
[tex]\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\][/tex]
### Step 2: Substitute Factored Forms Into Expression
Substituting the factored forms back into the original expression, we have:
[tex]\[
\frac{(x - 5)(x + 1)}{(x - 2)(x - 1)} \times \frac{(x - 3)(x - 2)}{(x - 5)(x + 2)}
\][/tex]
### Step 3: Cancel Common Factors
We can now cancel out common terms present in the numerator and the denominator:
1. (x - 5) appears once in both the numerator and the denominator, so we cancel them.
2. (x - 2) appears once in both the numerator and the denominator, so we also cancel them.
So, the resulting expression after cancelations will be:
[tex]\[
\frac{(x + 1) (x - 3)}{(x - 1) (x + 2)}
\][/tex]
### Step 4: Final Simplified Form
The simplified form of the expression is:
[tex]\[
\frac{(x - 3)(x + 1)}{(x - 1)(x + 2)}
\][/tex]
### Conclusion
In conclusion, the simplified form of the given expression:
[tex]\[
\frac{x^2 - 4x - 5}{x^2 - 3x + 2} \times \frac{x^2 - 5x + 6}{x^2 - 3x - 10}
\][/tex]
after factorization and cancellation of common factors, is:
[tex]\[
\frac{(x - 3)(x + 1)}{(x - 1)(x + 2)}
\][/tex]
