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a)
\(\begin{array}{l}\left( {2x - 5y} \right)\left( {2x + 5y} \right) + {\left( {2x + 5y} \right)^2}\\ = \left( {2x + 5y} \right)\left( {2x - 5y + 2x + 5y} \right)\\ = \left( {2x + 5y} \right).4x\\ = 2x.4x + 5y.4x\\ = 8{x^2} + 20xy\end{array}\)
b)
\(\begin{array}{l}\left( {x + 2y} \right)\left( {{x^2} - 2xy + 4{y^2}} \right) + \left( {2x - y} \right)\left( {4{x^2} + 2xy + {y^2}} \right)\\ = {x^3} + {\left( {2y} \right)^3} + {\left( {2x} \right)^3} - {y^3}\\ = {x^3} + 8{y^3} + 8{x^3} - {y^3}\\ = \left( {{x^3} + 8{x^3}} \right) + \left( {8{y^3} - {y^3}} \right)\\ = 9{x^3} + 7{y^3}\end{array}\)
\(A=x^2\left(x+y\right)+y^2\left(x+y\right)+2x^2y+2xy^2\)
\(=x^2\left(x+y\right)+y^2\left(x+y\right)+2xy\left(x+y\right)\)
\(=\left(x+y\right)\left(x^2+y^2+2xy\right)\)
\(=\left(x+y\right)\left(x+y\right)^2=\left(x+y\right)^3\)
\(A=x^2\left(x+y\right)+y^2\left(x+y\right)+2x^2y+2xy^2\)
\(A=x^2\left(x+y\right)+y^2\left(x+y\right)+2xy\left(x+y\right)\)
\(A=\left(x+y\right)\left(x^2+2xy+y^2\right)\)
\(A=\left(x+y\right)\left(x^2+2xy+y^2\right)\)
\(A=\left(x+y\right).\left(x+y\right)^2\)
\(A=\left(x+y\right)^3\)
a) \(\left(x-5\right)\left(a^2+5a+25\right)\)
\(=a^3-5^3\)
\(=a^3-125\)
b) \(\left(x+2y\right)\left(x^2-2xy+4y^2\right)\)
\(=x^3+\left(2y\right)^3\)
\(=x^3+8y^3\)
\(A=x^2\left(x+y\right)+y^2\left(x+y\right)+2xy\left(x+y\right)\)
\(\Leftrightarrow A=\left(x+y\right)\left(x^2+2xy+y^2\right)=\left(x+y\right)\left(x+y\right)^2=\left(x+y\right)^3\)
\(A=x^2\left(x+y\right)+y^2\left(x+y\right)+2x^2y+2xy^2\)
\(\Leftrightarrow A=\left(x^2+y^2\right)\left(x+y\right)+2xy\left(x+y\right)\)
\(\Leftrightarrow\left(x^2+2xy+y^2\right)\left(x+y\right)\)
\(\Leftrightarrow A=\left(x+y\right)^2\left(x+y\right)\)
\(\Leftrightarrow A=\left(x+y\right)^3\)
Check lại :v
\(\begin{array}{l}\left( {x - 2y} \right)\left( {{x^2} + 2xy + 4{y^2}} \right) + \left( {x + 2y} \right)\left( {{x^2} - 2xy + 4{y^2}} \right)\\ = {x^3} - {\left( {2y} \right)^3} + {x^3} + {\left( {2y} \right)^3}\\ = {x^3} - 8{y^3} + {x^3} + 8{y^3}\\ = 2{x^3}\end{array}\)