Sure, let's simplify the expression [tex]\(\frac{3x^2}{x+2} \cdot \frac{x^2 + 3x + 2}{6x^2 + 6x}\)[/tex] step-by-step.
First, let's rewrite the expression:
[tex]\[
\frac{3x^2}{x+2} \cdot \frac{x^2 + 3x + 2}{6x^2 + 6x}
\][/tex]
We'll start by factoring wherever possible:
1. The numerator of the second fraction [tex]\(x^2 + 3x + 2\)[/tex] can be factored:
[tex]\[
x^2 + 3x + 2 = (x + 1)(x + 2)
\][/tex]
2. The denominator of the second fraction [tex]\(6x^2 + 6x\)[/tex] can be factored by taking out the common factor [tex]\(6x\)[/tex]:
[tex]\[
6x^2 + 6x = 6x(x + 1)
\][/tex]
Substituting these factored forms back into the expression, we get:
[tex]\[
\frac{3x^2}{x+2} \cdot \frac{(x+1)(x+2)}{6x(x+1)}
\][/tex]
Next, we can cancel out the common factors in the numerator and the denominator:
- [tex]\(x+2\)[/tex] can be cancelled from the numerator and denominator:
[tex]\[
\frac{3x^2}{x+2} \cdot \frac{(x+1)(x+2)}{6x(x+1)} = \frac{3x^2}{1} \cdot \frac{(x+1)}{6x(x+1)} = \frac{3x^2 (x+1)}{6x (x+1) (x+2)/(x+2)} = \frac{3x^2}{6x}
\][/tex]
- [tex]\(x+1\)[/tex] is present in numerator and denominator, so it cancels out:
[tex]\[
\frac{3x^2}{6x}
\][/tex]
Finally, dividing the remaining factors:
[tex]\[
\frac{3x^2}{6x} = \frac{3}{6} \cdot \frac{x^2}{x} = \frac{1}{2} \cdot x = \frac{x}{2}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{x}{2}}
\][/tex]
