Factorise :
4x2+14x2+14x^2 + \dfrac{1}{4x^2} + 14x2+4x21+1
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Given,
=4x2+14x2+1=4x2+14x2+2−1=(2x)2+(12x)2+2×2x×12x−1=(2x+12x)2−12=(2x+12x+1)(2x+12x−1).\phantom{=} 4x^2 + \dfrac{1}{4x^2} + 1 \\[1em] = 4x^2 + \dfrac{1}{4x^2} + 2 - 1 \\[1em] = (2x)^2 + \Big(\dfrac{1}{2x}\Big)^2 + 2 \times 2x \times \dfrac{1}{2x} - 1 \\[1em] = \Big(2x + \dfrac{1}{2x}\Big)^2 - 1^2 \\[1em] = \Big(2x + \dfrac{1}{2x} + 1\Big)\Big(2x + \dfrac{1}{2x} - 1\Big).=4x2+4x21+1=4x2+4x21+2−1=(2x)2+(2x1)2+2×2x×2x1−1=(2x+2x1)2−12=(2x+2x1+1)(2x+2x1−1).
Hence, 4x2+14x2+1=(2x+12x+1)(2x+12x−1).4x^2 + \dfrac{1}{4x^2} + 1 = \Big(2x + \dfrac{1}{2x} + 1\Big)\Big(2x + \dfrac{1}{2x} - 1\Big).4x2+4x21+1=(2x+2x1+1)(2x+2x1−1).
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