Expand: $(a^{2} + 4 b^{2}) (a + 2 b) (a - 2 b)$

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Solution

Using the identity is $(x + y) (x - y) = x^{2} - y^{2}$ .

Given: $(a^{2} + 4 b^{2}) (a + 2 b) (a - 2 b)$ we apply the above identity in 2nd and 3rd term as follows:

$(a^{2} + 4 b^{2}) (a + 2 b) (a - 2 b) = (a^{2} + 4 b^{2}) [{(a)}^{2} - {(2 b)}^{2}] = (a^{2} + 4 b^{2}) (a^{2} - 4 b^{2})$

Again using the identity $(x + y) (x - y) = x^{2} - y^{2}$ :
$(a^{2} + 4 b^{2}) (a^{2} - 4 b^{2}) = {(a^{2})}^{2} - {(4 b^{2})}^{2} = a^{4} - 16 b^{4}$

Hence, $(a^{2} + 4 b^{2}) (a + 2 b) (a - 2 b) = a^{4} - 16 b^{4}$ .

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