To find the product of the given expression step-by-step, follow these steps:
1. Factor each polynomial expression in both numerators and denominators.
2. Rewrite the fractions with the factored forms.
3. Combine the fractions by multiplying them.
4. Simplify the combined fraction by canceling common factors.
Here's the detailed process in order:
1. Start by factoring each polynomial:
[tex]\[\frac{x^2 + 7x + 10}{x^2 + 4x + 4} \cdot \frac{x^2 + 3x + 2}{x^2 + 6x + 5}\][/tex]
2. Factor the polynomials:
[tex]\[x^2 + 7x + 10 = (x + 2)(x + 5)\][/tex]
[tex]\[x^2 + 4x + 4 = (x + 2)(x + 2)\][/tex]
[tex]\[x^2 + 3x + 2 = (x + 1)(x + 2)\][/tex]
[tex]\[x^2 + 6x + 5 = (x + 5)(x + 1)\][/tex]
3. Rewrite the fractions with the factored forms:
[tex]\[\frac{(x + 2)(x + 5)}{(x + 2)(x + 2)} \cdot \frac{(x + 1)(x + 2)}{(x + 5)(x + 1)}\][/tex]
4. Combine the fractions by multiplying them:
[tex]\[\frac{(x + 2)(x + 5)(x + 1)(x + 2)}{(x + 2)(x + 2)(x + 5)(x + 1)}\][/tex]
5. Cancel out the common factors from numerator and denominator:
[tex]\[\frac{\cancel{(x + 2)}\cancel{(x + 5)}\cancel{(x + 1)}\cancel{(x + 2)}}{\cancel{(x + 2)}\cancel{(x + 2)}\cancel{(x + 5)}\cancel{(x + 1)}}\][/tex]
After canceling the common factors, you are left with:
[tex]\[x + 2\][/tex]
Therefore, the correct steps are:
1. Factor each expression:
[tex]\[\frac{(x + 2)(x + 5)}{(x + 2)(x + 2)} \cdot \frac{(x + 1)(x + 2)}{(x + 5)(x + 1)}\][/tex]
2. Cancel the common factors and simplify:
[tex]\[x + 2\][/tex]
So, the correct sequence of steps utilizing the provided options is:
1. [tex]$\frac{(x+2)(x+5)}{(x+2)(x+2)} \cdot \frac{(x+1)(x+2)}{(x+5)(x+1)}$[/tex]
2. [tex]$x+2$[/tex]
