MHB How Do You Factor the Fractional Expression [(a + b)^2]/4 - [a^2b^2]/9?

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The expression [(a + b)^2]/4 - [a^2b^2]/9 can be factored as a difference of squares. It can be rewritten as (1/36)((3(a+b))^2 - (2ab)^2). By applying the difference of squares formula, further simplification can be achieved. This approach provides a clear pathway to factor the given fractional expression effectively. The discussion highlights the importance of recognizing patterns in algebraic expressions for easier manipulation.
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Factor [(a + b)^2]/4 - [a^2b^2]/9

Can someone get me started?
 
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I would write the expression as the difference of squares:

$$\frac{(a+b)^2}{4}-\frac{a^2b^2}{9}=\left(\frac{a+b}{2}\right)^2-\left(\frac{ab}{3}\right)^2$$

Or, we could write:

$$\frac{(a+b)^2}{4}-\frac{a^2b^2}{9}=\frac{1}{36}\left(\left(3(a+b)\right)^2-(2ab)^2\right)$$

Now apply the difference of squares formula...:D
 
Great. I can take it from here.
 

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