a) \(x^2+4y^2+4xy\)

\(=x^2+2.x.2y+\left(2y\right)^2\)

\(=\left(x+2y\right)^2\)

b) \(\left(x+y\right)^2-\left(x-y\right)^2\)

\(=\left(x+y-x+y\right)\left(x+y+x-y\right)\)

\(=2y.2x\)

\(=4xy\)

c) \(\left(3x+1\right)^2-\left(x+1\right)^2\)

\(=\left(3x+1-x-1\right)\left(3x+1+x-1\right)\)

a) \(x^6-y^6=\left(x^2\right)^3-\left(y^2\right)^3\)

\(=\left(x^2-y^2\right)\left(x^4+x^2y^2+y^4\right)\)

x² - 5x + 5

= x² - 5(x + 1)

= x² - \(\left[\sqrt{5\left(x+1\right)}\right]^2\)

= [x - \(\sqrt{5\left(x+1\right)}\)][x + \(\sqrt{5\left(x+1\right)}\)]

\(3x-7\sqrt{x}+4=3x-4\sqrt{x}-3\sqrt{x}+4=\sqrt{x}\left(3\sqrt{x}-4\right)-\left(3\sqrt{x}-4\right)=\left(3\sqrt{x}-4\right)\left(\sqrt{x}-1\right)\)

\(3x-7\sqrt{x}+4=3x-3\sqrt{x}-4\sqrt{x}+4\)

\(=3\sqrt{x}\left(\sqrt{x}-1\right)-4\left(\sqrt{x}-1\right)\)

\(=\left(3\sqrt{x}-4\right)\cdot\left(\sqrt{x}-1\right)\)

\(3x-7\sqrt{x}+4=3x-3\sqrt{x}-4\sqrt{x}+4=3.\left(\sqrt{x}\right)^2-3\sqrt{x}-4\sqrt{x}+4=3\sqrt{x}\left(\sqrt{x}-1\right)-4\left(\sqrt{x}-1\right)=\left(3\sqrt{x}-4\right)\left(\sqrt{x}-1\right)\)

\(x+\sqrt{x}-12=\left(\sqrt{x}+4\right)\left(\sqrt{x}-3\right)\)

\(2x-5\sqrt[]{x}+3=2x-2\sqrt[]{x}-3\sqrt[]{x}+3=2\sqrt[]{x}\left(\sqrt[]{x}-1\right)-3\sqrt[]{x}\left(\sqrt[]{x}-1\right)=\left(2\sqrt[]{x}-3\sqrt[]{x}\right)\left(\sqrt[]{x}-1\right)\)

\(x+\sqrt{x}-12=\left(\sqrt{x}+4\right)\left(\sqrt{x}-3\right)\)

\(x^2-y-y^2-x=\left(x-y\right)\left(x+y\right)-\left(x+y\right)=\left(x+y\right)\left(x-y-1\right)\)

x² - y - y² - x

= (x² - y²) - x - y

= (x - y)(x + y) - (x + y)

= (x - y - 1)(x + y)

\(=6x^2+5x-3xy\)

\(=x\left(6x+5-3y\right)\)

x⁸+64

=(x⁴)²+16x⁴+8²-16x⁴

=(x⁴+8)²-(4x²)²

=(x⁴+8-4x²)(x⁴+8+4x²)