Question

Find the sum.
5
2a
a? + 2a + 1
+
a2 + 4a + 3

Answer 1

`(2a)/(a^2+2a+1)+(5)/(a^2+4a+3)`

Factor the denominators.

`(2a)/(\left(a+1\right)^2)+(5)/(\left(a+1\right)\left(a+3\right))`

Adjust fractions based on LCM.

`(2a\left(a+3\right))/(\left(a+1\right)^2\left(a+3\right))+(5\left(a+1\right))/(\left(a+1\right)^2\left(a+3\right))`

Denominators are same, so add the fractions.

`(2a\left(a+3\right)+5\left(a+1\right))/(\left(a+1\right)^2\left(a+3\right))`

Expand the numerator.

`(2a^2+11a+5)/(\left(a+1\right)^2\left(a+3\right))`

Answer 2

`= \frac{2 {a}^(2) + 11a + 5}{ {(a + 1)}^(2)(a + 3)} \\`

Explanation:

`\frac{2a}{ {a}^(2) + 2a + 1} + \frac{5}{ {a}^(2) + 4a + 3 } \\ (2a)/((a + 1)(a + 1)) + (5)/((a + 1)(a + 3)) \\ \frac{2a}{ {(a + 1)}^(2) } + (5)/((a + 1)(a + 3)) \\ \frac{2a(a + 3) + 5(a + 1)}{ {(a + 1)}^(2)(a + 3) } \\ \frac{2 {a}^(2) + 6a + 5a + 5 }{ {(a + 1)}^(2)(a + 3)} \\ = \frac{2 {a}^(2) + 11a + 5}{ {(a + 1)}^(2)(a + 3)}`