Explanation:
The expression that we have is:
[tex]\frac{(x-1)^2}{x^2-x-12}\cdot\frac{x^2+x-6}{x^2-6x+5}[/tex]
And we need to find the equivalent expression.
Step 1. First, we expand the term (x-1)^2 as follows:
[tex]\frac{(x-1)(x-1)}{x^2-x-12}\cdot\frac{x^{2}+x-6}{x^{2}-6x+5}[/tex]
Then, we need to factor the rest of the three quadratic expressions.
Let's review the general process:
-For quadratic expressions of the form:
[tex]x^2+bx+c[/tex]
We factor it by finding two numbers that when you add them the result is b and when you multiply them the result is c.
Step 2. Factoring the three quadratic expressions:
[tex]x^2-x-12=(x-4)(x+3)[/tex][tex]x^2+x-6=(x-2)(x+3)[/tex][tex]x^2-6x+5=(x-5)(x-1)[/tex]
Step 3. Using the factored expressions, the result is:
We cancel the terms that are both in the numerator and in the denominator:
And we are left only with the following terms:
[tex]\frac{(x-1)(x-2)}{(x-4)(x-5)}[/tex]
Step 4. Multiplying the terms to find the final expression:
[tex]\frac{(x-1)(x-2)}{(x-4)(x-5)}=\frac{x^2-2x-x+2}{x^2-5x-4x+20}[/tex]
Combining like terms:
[tex]\frac{x^2-3x+2}{x^2-9x+20}[/tex]
This is shown in option A.
Answer:
[tex]\frac{x^{2}-3x+2}{x^{2}-9x+20}[/tex]
