a: \(\left(x^2-3\right)^2+16\)
\(=x^4-6x^2+9+16\)
\(=x^4-6x^2+25\)
\(=x^4+10x^2+25-16x^2\)
\(=\left(x^2+5\right)^2-\left(4x\right)^2=\left(x^2+5-4x\right)\left(x^2+5+4x\right)\)
b: \(\left(2x^2-4\right)^2+9\)
\(=4x_{}^4-16x^2+16+9\)
\(=4x^4-16x^2+25=4x^4+20x^2+25-36x^2\)
\(=\left(2x^2+5\right)^2-\left(6x\right)^2=\left(2x^2-6x+5\right)\left(2x^2+6x+5\right)\)
c: \(a^6+a^4+a^2b^2+b^4-b^6\)
\(=\left(a^6-b^6\right)+\left(a^4+a^2b^2+b^4\right)\)
\(=\left(a^2-b^2\right)\left(a^4+a^2b^2+b^4\right)+\left(a^4+a^2b^2+b^4\right)\)
\(=\left(a^2-b^2+1\right)\left(a^4+a^2b^2+b^4\right)\)
\(=\left(a^2-b^2+1\right)\left(a^4+2a^2b^2+b^4-a^2b^2\right)\)
\(=\left(a^2-b^2+1\right)\left\lbrack\left(a^2+b^2\right)^2-\left(ab\right)^2\right\rbrack=\left(a^2-b^2+1\right)\left(a^2+b^2-ab\right)\left(a^2+b^2+ab\right)\)
d: \(x^3+3xy+y^3-1\)
\(=\left(x+y\right)^3-3xy\left(x+y\right)+3xy-1\)
\(=\left\lbrack\left(x+y\right)^3-1\right\rbrack-3xy\left(x+y-1\right)\)
\(=\left(x+y-1\right)\left\lbrack\left(x+y\right)^2+x+y+1\right\rbrack-3xy\left(x+y-1\right)\)
\(=\left(x+y-1\right)\left(x^2+2xy+y^2+x+y+1-3xy\right)\)
\(=\left(x+y-1\right)\left(x^2+y^2-xy+x+y+1\right)\)
e: \(x^8+x_{}^7+1\)
\(=x^8+x^7+x^6-x^6-x^5-x^4+x^5+x^4+x^3-x^3-x^2-x+x^2+x+1\)
\(=x^6\left(x^2+x+1\right)-x^4\left(x^2+x+1\right)+x^3\left(x^2+x+1\right)-x\left(x^2+x+1\right)+\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^6-x^4+x^3-x+1\right)\)
f: \(x^7+x^5+1\)
\(=x^7+x^6+x^5-x^6-x^5-x^4+x^5+x^4+x^3-x^3+1\)
\(=x^5\left(x^2+x+1\right)-x^4\left(x^2+x+1\right)+x^3\left(x^2+x+1\right)-\left(x-1\right)\left(x^2+x+1\right)\)
\(=\left(x^2+x+1\right)\left(x^5-x^4+x^3-x+1\right)\)
h: \(x^5+x-1\)
\(=x^5-x^4+x^3+x^4-x^3+x^2-x^2+x-1\)
\(=x^3\left(x^2-x+1\right)+x^2\left(x^2-x+1\right)-\left(x^2-x+1\right)=\left(x^2-x+1\right)\left(x^3+x^2-1\right)\)
