a) \(4b^2c^2-\left(b^2+c^2-a^2\right)^2\)

\(=\left(2bc+b^2+c^2-a^2\right)\left(2bc-b^2-c^2+a^2\right)\)

\(=\left[\left(b+c\right)^2-a^2\right]\left[a^2-\left(b-c\right)^2\right]\)

\(=\left(b+c+a\right)\left(b+c-a\right)\left(a+b-c\right)\left(a-b+c\right)\)

b) \(\left(ax+by\right)^2-\left(ay+bx\right)^2\)

\(=\left(ax+by+ay+bx\right)\left(ax+by-ay-bx\right)\)

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

c) \(\left(a^2+b^2-5\right)^2-4\left(ab+2\right)^2\)

\(=\left(a^2+b^2-5+2ab+4\right)\left(a^2+b^2-5-2ab-4\right)\)

\(=\left[\left(a+b\right)^2-1\right]\left[\left(a-b\right)^2-9\right]\)

\(=\left(a+b+1\right)\left(a+b-1\right)\left(a-b+3\right)\left(a-b-3\right)\)

d) \(\left(4x^2-3x-18\right)^2-\left(4x^2+3x\right)^2\)

\(=\left(4x^2-3x-18+4x^2+3x\right)\left(4x^2-3x-18-4x^2-3x\right)\)

\(=\left(8x^2-18\right)\left(-6x-18\right)\)

\(=\left[2\left(4x^2-9\right)\right]\left[-6\left(x+3\right)\right]\)

\(=12\left(2x+3\right)\left(2x-3\right)\left(x+3\right)\)

a) \(4x^2-1=\left(2x+1\right)\left(2x-1\right)\)

b) \(\left(x+2\right)^2-9=\left(x-1\right)\left(x+5\right)\)

c) \(\left(a+b\right)^2-\left(a-2b\right)^2\)

\(=\left(a+b-a+2b\right)\left(a+b+a-2b\right)\)

\(=3b\left(2a-b\right)\)

`a, 4x^2-1 = (2x+1)(2x-1)`

`b, (x+2)^2-9 = (x+2-3)(x+2+3) = (x-1)(x+5)`

`c, (a+b)^2-(a-2b)^2 = (a+b+a-2b)(a+b-a+2b) = (2a-b)(3b)`

a: \(x^2-2xy+y^2+3x-3y-4\)

\(=\left(x-y\right)^2+3\left(x-y\right)-4\)

\(=\left(x-y+4\right)\left(x-y-1\right)\)

`a, 4a^2 + 4a + 1 = (2a+1)^2`

`b, -3x^2 + 6xy - 3y^2`

` = -3(x-y)^2`

`c, (x+y)^2 - 2(x+y)z + z^2`

`= (x+y-z)^2`

`a, x^3 + 4x = x(x^2+4)`

`b, 6ab - 9ab^2 = 3ab(2-b)`

`c, 2a(x-1) + 3b(1-x)`

`= (2a-3b)(x-1)`

`d, (x-y)^2 - x(y-x)`

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

`= (2x-y)(x-y)`

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