To solve the given problem, we need to determine the product of the two fractions [tex]\(\frac{x^2+7x+10}{x^2+4x+4} \cdot \frac{x^2+3x+2}{x^2+6x+5}\)[/tex] in a step-by-step manner.
Here's the correct order of steps:
1. Factor the numerators and denominators.
The given fractions are:
[tex]\(\frac{x^2 + 7x + 10}{x^2 + 4x + 4} \cdot \frac{x^2 + 3x + 2}{x^2 + 6x + 5}\)[/tex]
After factoring, we get:
[tex]\(\frac{(x+5)(x+2)}{(x+2)(x+2)} \cdot \frac{(x+1)(x+2)}{(x+5)(x+1)}\)[/tex]
2. Express the product of the fractions with the factored forms:
[tex]\(\frac{(x+5)(x+2)}{(x+2)(x+2)} \cdot \frac{(x+1)(x+2)}{(x+5)(x+1)}\)[/tex]
3. Simplify the product by canceling out common factors:
After canceling out [tex]\((x+2)\)[/tex] and other possible common factors, the expression simplifies to:
[tex]\(\frac{(x+5)(x+2)}{(x+2)(x+2)} \cdot \frac{(x+1)(x+2)}{(x+5)(x+1)}\)[/tex]
This simplifies to:
[tex]\(\frac{(x+5)}{(x+2)} \cdot \frac{(x+2)}{(x+5)}\)[/tex]
4. Further simplify:
Simplifying further gives:
[tex]\(\frac{(x+5)}{(x+2)} \cdot \frac{(x+2)}{(x+5)} = 1\)[/tex]
The correct steps in sequence are:
1. [tex]\(\frac{(x+5)(x+2)}{(x+2)(x+2)} \cdot \frac{(x+1)(x+2)}{(x+5)(x+1)}\)[/tex]
2. [tex]\(\frac{(x+5)}{(x+2)} \cdot \frac{(x+2)}{(x+5)}\)[/tex]
Selecting the relevant tiles:
1. [tex]\(\frac{(x+5)(x+2)}{(x+2)(x+2)} \cdot \frac{(x+1)(x+2)}{(x+5)(x+1)}\)[/tex]
2. [tex]\(\frac{(x+5)}{(x+2)} \cdot \frac{(x+2)}{(x+5)}\)[/tex]
