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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: \(x^2-2xy+y^2+3x-3y-4\)
\(=\left(x-y\right)^2+3\left(x-y\right)-4\)
\(=\left(x-y+4\right)\left(x-y-1\right)\)
\(\left(ax+by\right)^2-\left(ay+bx\right)^2\)
\(=\left(ax+by+ay+bx\right)\left(ax+by-ay-bx\right)\)
\(=\left[a\left(x+y\right)+b\left(x+y\right)\right]\left[a\left(x-y\right)-b\left(x-y\right)\right]\)
\(=\left(a+b\right)\left(a-b\right)\left(x+y\right)\left(x-y\right)\)
\(\left(a^2+b^2-5\right)^2-4\left(ab+2\right)^2\)
\(=\left[\left(a^2+b^2-5\right)+2\left(ab+2\right)\right]\left[\left(a^2+b^2-5\right)-2\left(ab+2\right)\right]\)
\(=\left[a^2+b^2-5+2ab+4\right]\left[a^2+b^2-5-2ab-4\right]\)
\(=\left[\left(a+b\right)^2-1\right]\left[\left(a-b\right)^2-9\right]\)
\(=\left(a+b-1\right)\left(a+b+1\right)\left(a-b-3\right)\left(a-b+3\right)\)
a)
(ax+by)2 - (ay+bx)2
=(ax+by-ay-bx)(ax+by+ay+bx)
=[ a(x-y) -b(x-y)][ a(x+y) + b(x+y)]
=(a-b)(x-y)(a+b)(x+y)
b)(a2+b2-5)2 - 4(ab+2)2
=(a2+b2-5-2ab-4)(a2+b2-5+2ab+4)
=[ (a-b)2 -9][ (a+b)2 -1]
=(a-b-3)(a-b+3)(a+b-1)(a+b+1)