Sure! Let's simplify the given expression step-by-step:
The given expression is:
[tex]\[ \frac{1}{a + 2b} + \frac{2a}{a^2 - 4b^2} \][/tex]
First, observe that the denominator of the second term can be factored using the difference of squares:
[tex]\[ a^2 - 4b^2 = (a - 2b)(a + 2b) \][/tex]
So the expression becomes:
[tex]\[ \frac{1}{a + 2b} + \frac{2a}{(a - 2b)(a + 2b)} \][/tex]
Next, we want to combine these two fractions into a single fraction. To do this, we need a common denominator. The common denominator here is:
[tex]\[ (a + 2b)(a - 2b) \][/tex]
Rewriting each part with the common denominator, we get:
[tex]\[ \frac{1 \cdot (a - 2b)}{(a + 2b)(a - 2b)} + \frac{2a}{(a + 2b)(a - 2b)} \][/tex]
Now, we have:
[tex]\[ \frac{a - 2b}{(a + 2b)(a - 2b)} + \frac{2a}{(a + 2b)(a - 2b)} \][/tex]
Since the denominators are the same, we can combine the numerators:
[tex]\[ \frac{(a - 2b) + 2a}{(a + 2b)(a - 2b)} \][/tex]
Simplifying the numerator:
[tex]\[ a - 2b + 2a = 3a - 2b \][/tex]
So the expression now looks like:
[tex]\[ \frac{3a - 2b}{(a + 2b)(a - 2b)} \][/tex]
Thus, the simplified form of the expression is:
[tex]\[ \frac{3a - 2b}{a^2 - 4b^2} \][/tex]
Therefore, the simplified expression is:
[tex]\[ \boxed{\frac{3a - 2b}{a^2 - 4b^2}} \][/tex]
