Question

Factorise:
4a² – 4b² + 4a + 1

Answer 1

$$\begin{gathered} 4a^2-4b^2+4a+1 \\ =\left(4a^2+4a+1\right)-4b^2 \\ =\left[{\left(2a\right)}^2+2\times 2a\times 1+1^2\right]-4b^2 \\ ={\left(2a+1\right)}^2-{\left(2b\right)}^2\left[a^2+2ab+b^2={\left(a+b\right)}^2\right] \end{gathered}$$

$$\begin{gathered} =\left[\left(2a+1\right)+2b\right]\left[\left(2a+1\right)-2b\right]\left[a^2-b^2=\left(a+b\right)\left(a-b\right)\right] \\ =\left(2a+1+2b\right)\left(2a+1-2b\right) \\ =\left(2a+2b+1\right)\left(2a-2b+1\right) \end{gathered}$$