a) \(\left(3x-1\right)^2-16\)
\(=\left(3x-1\right)^2-4^2\)
\(=\left(3x-1+4\right)\left(3x-1-4\right)\)
\(=\left(3x+3\right)\left(3x-5\right)\)
b) \(\left(3x+1\right)^2-4\left(x-2\right)^2\)
\(=\left(3x+1\right)^2-\left[2\left(x-2\right)\right]^2\)
\(=\left(3x+1\right)^2-\left(2x-4\right)^2\)
\(=\left[\left(3x+1\right)+\left(2x-4\right)\right]\left[\left(3x+1\right)-\left(2x-4\right)\right]\)
\(=\left(5x-3\right)\left(x+5\right)\)
c) \(\left(ax+by\right)^2-\left(ax+by\right)^2\)
\(=\left[\left(ax+by\right)+\left(ax+by\right)\right]\left[\left(ax+by\right)-\left(ax-by\right)\right]\)
\(=\left(2ax+2by\right)\cdot0\)
\(=0\)
d) \(\left(4x^2-3x-18\right)^2-\left(4x^2+3x\right)^2\)
\(=\left(4x^2-3x-18+4x^2+3x\right)\left(4x^2-3x-18-4x^2-3x\right)\)
\(=\left(8x^2-18\right)\left(-6x-18\right)\)
e) \(-4x^2+12xy-9y^2+25\)
\(=-\left(4x^2-12xy+9y^2-25\right)\)
\(=-\left[\left(2x+3y\right)^2-5^2\right]\)
\(=-\left(2x+3y+5\right)\left(2x+3y-5\right)\)
g) \(x^3+6x^2+12x+8\)
\(=\left(x+8\right)^3\)
h) \(x^3+\dfrac{3}{2}x^2+\dfrac{3}{4}x+\dfrac{1}{8}\)
\(=\left(x+\dfrac{1}{2}\right)^3\)
