Let's go through the solution step-by-step to determine the product of the given fractions.
1. Factorize the numerators and denominators of both fractions:
The original fractions are:
[tex]\[
\frac{x^2 + 7x + 10}{x^2 + 4x + 4} \cdot \frac{x^2 + 3x + 2}{x^2 + 6x + 5}
\][/tex]
Factoring them, we get:
[tex]\[
\frac{(x + 5)(x + 2)}{(x + 2)(x + 2)} \cdot \frac{(x + 2)(x + 1)}{(x + 5)(x + 1)}
\][/tex]
So the first step is:
[tex]\[
\frac{(x+2)(x+5)}{(x+2)(x+2)} \cdot \frac{(x+1)(x+2)}{(x+5)(x+1)}
\][/tex]
2. Combine the factors in the numerators and denominators:
When combined, it becomes:
[tex]\[
\frac{(x + 5)(x + 2)(x + 2)(x + 1)}{(x + 2)(x + 2)(x + 5)(x + 1)}
\][/tex]
3. Cancel out the common factors:
The common factors in the numerator and the denominator are [tex]\((x + 5)\)[/tex], [tex]\((x + 2)\)[/tex], and [tex]\((x + 1)\)[/tex]. Canceling these out, we are left with:
[tex]\[
\frac{1}{1}
\][/tex]
4. Simplify the result:
The final simplified product is:
[tex]\[
1
\][/tex]
So, the correct order in which these steps should be performed is as follows:
1. [tex]\(\frac{(x+2)(x+5)}{(x+2)(x+2)} \cdot \frac{(x+1)(x+2)}{(x+5)(x+1)}\)[/tex]
2. [tex]\(\frac{(x+5)(x+2)(x+2)(x+2)(x+1)}{(x+2)(x+2)(x+5)(x+1)}\)[/tex]
3. Cancel the common factors.
4. Simplify to get [tex]\(1\)[/tex].
