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\(=\left(x+2\right)\left(x+5\right)\left(x+3\right)\left(x+4\right)-24\)
\(=\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\)
đặt \(t=x^2+7x+10\Rightarrow x^2+7x+12=t+2\)
\(\Rightarrow\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24=t\left(t+2\right)-24=t^2+2t-24=\left(t-4\right)\left(t+6\right)=\)
\(=\left(x^2+7x+10-4\right)\left(x^2+7x+10+6\right)=\left(x^2+7x+6\right)\left(x^2+7x+16\right)\)
\(x^2+2xy+7x+7y+y^2+10\)
\(=\left(x^2+2xy+y^2\right)+\left(7x+7y\right)+\frac{49}{4}-\frac{9}{4}\)
\(=\left(x+y\right)^2+7\left(x+y\right)+\frac{49}{4}-\frac{9}{4}\)
\(=\left(x+y+\frac{7}{2}\right)^2-\frac{9}{4}\)
\(=\left(x+y+\frac{7}{2}-\frac{3}{2}\right)\left(x+y+\frac{7}{2}+\frac{3}{2}\right)\)
\(=\left(x+y+2\right)\left(x+y+5\right)\)
b)Ta có: x2y+xy2+x+y=2010
<=>xy.x+xy.y+x+y=2010
<=>11x+11y+x+y=2010
<=>12(x+y)=2010
<=>x+y=167,5
=>(x+y)2=28056,25
<=>x2+y2+2xy=28056,25
<=>x2+y2=28034,25
\(64x^4+y^4\)
\(=\left(64x^4+16x^2y^2+y^4\right)-16x^2y^2\)
\(=\left(8x^2+y^2\right)^2-16x^2y^2\)
\(=\left(8x^2+y^2-4xy\right)\left(8x^2+y^2+4xy\right)\)
\(x^5+x-1\)
\(=x^5+x^2-\left(x^2-x+1\right)\)
\(=x^2\left(x^3+1\right)-\left(x^2-x+1\right)\)
\(=x^2\left(x+1\right)\left(x^2-x+1\right)-\left(x^2-x+1\right)\)
\(=\left(x^2-x+1\right)\left(x^3+x^2-1\right)\)
\(x^2-3\)
\(=x^2-\left(\sqrt{3}\right)^2\)
\(=\left(x-\sqrt{3}\right)\left(x+\sqrt{3}\right)\)
\(x^{16}-1\)
\(=x^{16}+x^8-x^8-1\)
\(=x^8\left(x^8+1\right)-\left(x^8+1\right)\)
\(=\left(x^8+1\right)\left(x^8-1\right)\)
\(=\left(x^8+1\right)\left(x^8-x^4+x^4-1\right)\)
\(=\left(x^8+1\right)\left[x^4\left(x^4-1\right)+\left(x^4-1\right)\right]\)
\(=\left(x^8+1\right)\left(x^4+1\right)\left(x^4-1\right)\)
\(=\left(x^8+1\right)\left(x^4+1\right)\left(x^4-x^2+x^2-1\right)\)
\(=\left(x^8+1\right)\left(x^4+1\right)\left[x^2\left(x^2-1\right)+\left(x^2-1\right)\right]\)
\(=\left(x^8+1\right)\left(x^4+1\right)\left(x^2+1\right)\left(x^2-1\right)\)
\(=\left(x^8+1\right)\left(x^4+1\right)\left(x^2+1\right)\left(x^2-x+x-1\right)\)
\(=\left(x^8+1\right)\left(x^4+1\right)\left(x^2+1\right)\left[x\left(x-1\right)+\left(x-1\right)\right]\)
\(=\left(x^8+1\right)\left(x^4+1\right)\left(x^2+1\right)\left(x+1\right)\left(x-1\right)\)