Question
Factorise $: 4 x^2+\frac{1}{(4 x)^2}+1$
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Answer
$ 4 x^2+\frac{1}{(4 x)^2}+1$
$ =4 x^2+\frac{1}{(4 x)^2}+2-1$
$ =4 x^2+\frac{1}{(4 x)^2}+2 \times 2 x \times \frac{1}{2 x}-1$
$ =\left(2 x+\frac{1}{2 x}+1\right)^2-(1)^2$
$ =\left(2 x+\frac{1}{2 x}+1\right)\left(2 x+\frac{1}{2 x}-1\right)$
$ =4 x^2+\frac{1}{(4 x)^2}+2-1$
$ =4 x^2+\frac{1}{(4 x)^2}+2 \times 2 x \times \frac{1}{2 x}-1$
$ =\left(2 x+\frac{1}{2 x}+1\right)^2-(1)^2$
$ =\left(2 x+\frac{1}{2 x}+1\right)\left(2 x+\frac{1}{2 x}-1\right)$
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