1: \(\left(x+1\right)^2-\left(x-1\right)^2\)
\(=\left(x+1-x+1\right)\left(x+1+x-1\right)\)
\(=2\cdot2x=4x\)
2: \(\left(x+2\right)^2-2\left(x+2\right)\cdot x+x^2\)
\(=\left(x+2-x\right)^2=2^2=4\)
3: \(\left(x-3\right)^2+\left(x+3\right)^2\)
\(=x^2-6x+9+x^2+6x+9\)
\(=2x^2+18\)
4: \(\left(2x+1\right)\left(2x-1\right)=\left(2x\right)^2-1^2=4x^2-1\)
5: \(\left(y+2\right)^2-\left(y-2\right)^2\)
\(=y^2+4y+4-\left(y^2-4y+4\right)\)
\(=y^2+4y+4-y^2+4y-4=8y\)
6: \(\left(x+4\right)^2-2\left(x+4\right)\left(x-4\right)+\left(x-4\right)^2\)
\(=\left(x+4-x+4\right)^2\)
\(=8^2=64\)
7: \(\left(m+1\right)\left(m-1\right)+\left(m+1\right)^2\)
\(=m^2-1+m^2+2m+1\)
\(=2m^2+2m\)
8: \(\left(n+2\right)\left(n-2\right)-\left(n^2-4\right)\)
\(=n^2-4-\left(n^2-4\right)\)
=0
9: \(\left(a+b\right)^2-\left(a-b\right)^2\)
=(a+b+a-b)(a+b-a+b)
=2a*2b=4ab
10: \(\left(p+1\right)^2+\left(p-1\right)^2\)
\(=p^2+2p+1+p^2-2p+1=2p^2+2\)
11: \(\left(x+2\right)^2-2\left(x+2\right)\left(x-2\right)+\left(x-2\right)^2\)
\(=\left(x+2-x+2\right)^2=4^2=16\)
12: \(\left(x-3\right)^2+2\left(x-3\right)\left(x+3\right)+\left(x+3\right)^2\)
\(=\left(x-3+x+3\right)^2=\left(2x\right)^2=4x^2\)
13: \(\left(x-2\right)\cdot\left(x+2\right)\left(x^2+4\right)-\left(x^2-4\right)^2\)
\(=\left(x^2-4\right)\left(x^2+4\right)-\left(x^2-4\right)^2\)
\(=\left(x^2-4\right)\left(x^2+4-x^2+4\right)=8\left(x^2-4\right)=8x^2-32\)
14: \(\left(2x-3\right)\left(2x+3\right)\left(4x^2+9\right)-\left(4x^2-9\right)^2\)
\(=\left(4x^2-9\right)\left(4x^2+9\right)-\left(4x^2-9\right)^2\)
\(=\left(4x^2-9\right)\left(4x^2+9-4x^2+9\right)=18\left(4x^2-9\right)=72x^2-162\)
15: \(\left(4x+1\right)^2-\left(4x-1\right)^2\)
\(=\left(4x+1+4x-1\right)\left(4x+1-4x+1\right)\)
\(=8x\cdot2=16x\)
16: \(\left(x^2+2\right)^2-\left\lbrack\left(x-1\right)\left(x+1\right)\right\rbrack^2\)
\(=\left(x^2+2\right)^2-\left(x^2-1\right)^2\)
\(=\left(x^2+2+x^2-1\right)\left(x^2+2-x^2+1\right)=3\left(2x^2+1\right)=6x^2+3\)
17: \(\left(3x+1\right)^2-2\left(3x+1\right)\left(3x-1\right)+\left(3x-1\right)^2\)
\(=\left(3x+1-3x+1\right)^2=2^2=4\)
18: \(\left(x+1\right)\left(x-1\right)\left(x^2+1\right)+\left(x^2-1\right)^2\)
\(=\left(x^2-1\right)\left(x^2+1\right)+\left(x^2-1\right)^2\)
\(=\left(x^2-1\right)\cdot\left(x^2+1+x^2-1\right)=2x^2\left(x^2-1\right)=2x^4-2x^2\)
19: \(\left(x^2+3x+2\right)^2-\left(x^2-3x+2\right)^2\)
\(=\left(x^2+3x+2+x^2-3x+2\right)\left(x^2+3x+2-x^2+3x-2\right)\)
\(=6x\left(2x^2+4\right)=12x^3+24x\)
20: \(\left(x+y\right)\left(x-y\right)\left(x^2+y^2\right)-\left(x^2-y^2\right)^2\)
\(=\left(x^2-y^2\right)\left(x^2+y^2\right)-\left(x^2-y^2\right)^2\)
\(=\left(x^2-y^2\right)\left(x^2+y^2-x^2+y^2\right)=2y^2\left(x^2-y^2\right)=2x^2y^2-2y^4\)
