Certainly! Let's solve the problem step-by-step by multiplying and simplifying the given rational expressions:
[tex]\[
\frac{x^2 - 4x + 3}{x^2 + 3x + 2} \cdot \frac{x + 1}{3x - 9}
\][/tex]
Step 1: Factor each polynomial in the expressions, if possible.
1. [tex]\( x^2 - 4x + 3 \)[/tex] factors into:
[tex]\[ (x - 1)(x - 3) \][/tex]
since [tex]\( (x - 1)(x - 3) = x^2 - 4x + 3 \)[/tex].
2. [tex]\( x^2 + 3x + 2 \)[/tex] factors into:
[tex]\[ (x + 1)(x + 2) \][/tex]
since [tex]\( (x + 1)(x + 2) = x^2 + 3x + 2 \)[/tex].
3. [tex]\( x + 1 \)[/tex] does not need to be factored, as it is already a linear polynomial.
4. [tex]\( 3x - 9 \)[/tex] factors into:
[tex]\[ 3(x - 3) \][/tex]
since [tex]\( 3(x - 3) = 3x - 9 \)[/tex].
Step 2: Replace the original polynomial expressions with their factored forms.
[tex]\[
\frac{(x - 1)(x - 3)}{(x + 1)(x + 2)} \cdot \frac{x + 1}{3(x - 3)}
\][/tex]
Step 3: Combine the fractions by multiplying the numerators and the denominators.
Numerator:
[tex]\[
(x - 1)(x - 3)(x + 1)
\][/tex]
Denominator:
[tex]\[
(x + 1)(x + 2)3(x - 3)
\][/tex]
Step 4: Simplify by canceling out common factors from the numerator and the denominator.
1. [tex]\( (x + 1) \)[/tex] in the numerator and denominator cancel out.
2. [tex]\( (x - 3) \)[/tex] in the numerator and the denominator cancel out.
After canceling the common factors, we are left with:
[tex]\[
\frac{x - 1}{3(x + 2)}
\][/tex]
Thus, the simplified form of the expression is:
[tex]\[
\frac{x - 1}{3(x + 2)}
\][/tex]
This is the final simplified result.
