Final answer:
To find the sum of the expression 3n/ (n² + 4n + 4) + 7/(n² + 6n + 8), we need to combine the two terms together by finding a common denominator and then adding the numerators, we get 10n²+40n+28/(n + 2)². (n + 4).
Step-by-step explanation:
To find the sum of the expression 3n/ (n² + 4n + 4) + 7/(n² + 6n + 8), we need to combine the two terms together by finding a common denominator and then adding the numerators.
First, we factorize the denominators to get n² + 4n + 4 = (n + 2)² and n² + 6n + 8 = (n + 2). (n + 4).
Now rewrite the expression with the factored denominators:
3n/(n + 2)² +7/(n + 2)(n + 4)
To add these fractions, find a common denominator, which is the product of the two distinct factors: (n + 2)²(n + 4).
Now, adjust the numerators accordingly:
3n(n + 4)/(n + 2)². (n + 4) + 7(n + 2)²/ (n + 2)². (n + 4)
Combine the numerators over the common denominator:
3n(n + 4) + 7(n + 2)²/ (n + 2)². (n + 4)
Expand and simplify the numerator:
3n²+12n+7n²+28n+28/(n + 2)². (n + 4)
Combine like terms:
10n²+40n+28/(n + 2)². (n + 4)
So, the simplified expression is:
10n²+40n+28/(n + 2)². (n + 4)