%5E2%2B2(a%2B4)%2B1%2Fa%2B5.jpeg "(a+4)^2+2(a+4)+1/a+5")
Simplifying the Expression: (a + 4)^2 + 2(a + 4) + 1 / (a + 5)
This expression combines several algebraic concepts, including:
- Squaring a binomial: (a + 4)^2
- Distributing: 2(a + 4)
- Adding fractions: 1 / (a + 5)
Let's break down each part and simplify the entire expression:
Simplifying (a + 4)^2
We can use the formula for squaring a binomial: (x + y)^2 = x^2 + 2xy + y^2
Applying this to our case:
(a + 4)^2 = a^2 + 2(a)(4) + 4^2 = a^2 + 8a + 16
Simplifying 2(a + 4)
Using the distributive property:
2(a + 4) = 2a + 8
Combining the Terms
Now we can combine all the simplified parts:
(a + 4)^2 + 2(a + 4) + 1 / (a + 5) = a^2 + 8a + 16 + 2a + 8 + 1 / (a + 5)
Simplifying Further
Combine like terms:
a^2 + 10a + 24 + 1 / (a + 5)
To express this as a single fraction, we need to find a common denominator:
(a^2 + 10a + 24)(a + 5) / (a + 5) + 1 / (a + 5)
Now we can combine the numerators:
(a^2 + 10a + 24)(a + 5) + 1 / (a + 5)
Final Result
The simplified expression is:
(a^2 + 10a + 24)(a + 5) + 1 / (a + 5)
Important Note: This expression is undefined when a = -5, as it would result in division by zero.