$$\begin{gathered} 1, {\left(x+y+z\right)}^2-2\left(x+y+z\right)\left(y+z\right)+{\left(y+z\right)}^2 \\ ={\left(x+y+z-y-z\right)}^2 \\ =x^2 \\ 2, {\left(x+3\right)}^2+4\left(x+3\right)+4={\left(x+3+2\right)}^2={\left(x+5\right)}^2 \\ 3, {\left(x+2\right)}^2+2\left(x+2\right)\left(x-2\right)+{\left(x-2\right)}^2=\left(x+2+x-2\right)={\left(2x\right)}^2=4x^2 \\ 4, {\left(x+y+z\right)}^2-{\left(y+z\right)}^2 \\ =\left(x+y+z-y-z\right)\left(x+y+z+y+z\right) \\ =x.\left(x+2y+2z\right) \\ 5, 25+10\left(x+1\right)+{\left(x+1\right)}^2={\left(x+1\right)}^2+2.\left(x+1\right).5+5^2 \\ ={\left(x+1+5\right)}^2={\left(x+6\right)}^2 \\ 6, {\left(x-3\right)}^2-2.\left(x^2-9\right)+{\left(x+3\right)}^2 \\ ={\left(x-3\right)}^2-2.\left(x-3\right)\left(x+3\right)+{\left(x+3\right)}^2 \\ =\left(x-3-x-3\right)={\left(-6\right)}^2=36 \end{gathered}$$.

\[\begin{array}{l}
{\left( {x + y + z} \right)^2}\; - \;2\left( {x + y + z} \right)\left( {y + z} \right)\; + \;{\left( {y + z} \right)^2}\\
= {[\left( {x + y + z} \right) - \left( {y + z} \right)]^2}\\
= {\left( {x + y + z - y - z} \right)^2}\\
= {x^2}\\
{\left( {x + 3} \right)^2}\; + \;4\left( {x + 3} \right)\; + \;4\\
= {[\left( {x + 3} \right) + 2]^2}\\
= {\left( {x + 5} \right)^2}\\
{\left( {x + 2} \right)^2}\; + \;2\left( {x + 2} \right)\left( {x - 2} \right)\; + \;{\left( {x - 2} \right)^2}\\
= {[\left( {x + 2} \right) + \left( {x - 2} \right)]^2}\\
= {\left( {x + 2 + x - 2} \right)^2}\\
= {\left( {2x} \right)^2}\\
= 4{x^2}\\
{\left( {x + y + z} \right)^2}\; - \;{\left( {y + z} \right)^2}\\
= [\left( {x + y + z} \right) - \left( {y + z} \right)].[\left( {x + y + z} \right) + \left( {y + z} \right)]\\
= \left( {x + y + z - y - z} \right)\left( {x + y + z + y + z} \right)\\
= x.\left( {x + 2y + 2z} \right)\\
25 + 10\left( {\;x + 1\;} \right)\; + \;{\left( {x + 1} \right)^2}\\
= {\left( {5 + x + 1} \right)^2}\\
= {\left( {x + 6} \right)^2}\\
{\left( {x - 3} \right)^2}\; - \;2\left( {\;{x^2} - 9\;} \right)\; + \;{\left( {x + 3} \right)^2}\\
= {\left( {x - 3} \right)^2} - 2\left( {x - 3} \right)\left( {x + 3} \right) + {\left( {x + 3} \right)^2}\\
= {[\left( {x - 3} \right) - \left( {x + 3} \right)]^2}\\
= {\left( {x - 3 - x - 3} \right)^2}\\
= {\left( { - 6} \right)^2} = 36
\end{array}\]

HD

$$\begin{gathered} 1. {(x+y+z)}^2-2(x+y+z)(y+z)+{(y+z)}^2={\left[\right(x+y+z)-(y+z\left)\right]}^2 =(x+y+z-y-z{)}^2=x^2 \\ 2. {(x+3)}^2+4(x+3)+4={(x+3)}^2+2.2.(x+3)+2^2={(x+3+2)}^2={(x+5)}^2 3. {(x+2)}^2+2(x+2\left)\right(x-2)+{(x-2)}^2={(x+2+x-2)}^2=(2x{)}^2=4x^2 \\ 4. {(x+y+z)}^2-{(y+z)}^2=(x+y+z-y-z)(x+y+z+y+z)=x.(x+2y+2z) \\ 5. 25+10(x+1)+{(x+1)}^2=5^2+2.5.(x+1)+{(x+1)}^2={(x+1+5)}^2={(x+6)}^2 \\ 6. {(x-3)}^2-2(x^2-9)+{(x+3)}^2={(x-3)}^2-2(x-3)(x+3)+{(x+3)}^2 ={\left[\right(x-3)-(x+3\left)\right]}^2=(x-3-x-3{)}^2={(-6)}^2=36 \end{gathered}$$