Example  1
Factor the following differences of squares.

  1. \(4x^2 - 81y^2\)

    \(4x^2 - 81y^2 = \left( {2x + 9y} \right)\left( {2x - 9y} \right)\)

  2. \(\frac{1}{4}x^2 - 25y^4\)

    \[\frac{1}{4}x^2 - 25y^4 = \left( {\frac{1}{2}x + 5y^2 } \right)\left( {\frac{1}{2}x - 5y^2 } \right)\]


  3. \(16\left( {x^2 + 1} \right)^2 - 1\)

    The square root of the first term is \(4\left( {x^2 + 1} \right)\) and the square root of the last is \(1\).

    \(\begin{align}
     16\left( {x^2 + 1} \right)^2 - 1 &= \left[ {4\left( {x^2 + 1} \right) + 1} \right]\left[ {4\left( {x^2 + 1} \right) - 1} \right] \\
      &= \left( {4x^2 + 4 + 1} \right)\left( {4x^2 + 4 - 1} \right) \\
      &= \left( {4x^2 + 5} \right)\left( {4x^2 + 3} \right) \\
     \end{align}\)