Answer:
[tex]\qquad \:\boxed{\begin{aligned}& \qquad \:\sf \:(1) \: \: 3{(2a - 3b)} \: ( 8a - 12b- 5)\qquad \: \\ \\& \qquad \:\sf \:(2) \: \: (2m-5n)(2m - 5n - 5) \\ \\& \qquad \:\sf \:(3) \: \: 3(x-2y) \: (x - 2y - 2)\end{aligned}} \qquad \\ \\ [/tex]
Step-by-step explanation:
[tex]\large\underline{\sf{Solution-1}}[/tex]
Given algebraic expression is
[tex]\sf \: 12 {(2a - 3b)}^{2} - 15(2a - 3b) \\ \\ [/tex]
[tex]\qquad\sf \: = \: 2.2.3. {(2a - 3b)}^{2} - 3.5.(2a - 3b) \\ \\ [/tex]
[tex]\qquad\sf \: = \: 3{(2a - 3b)}[ 4(2a - 3b)- 5] \\ \\ [/tex]
[tex]\qquad\sf \: = \: 3{(2a - 3b)} \: ( 8a - 12b- 5) \\ \\ [/tex]
Hence,
[tex]\boxed{ \sf{ \:12 {(2a - 3b)}^{2} - 15(2a - 3b) = 3{(2a - 3b)} \: ( 8a - 12b- 5) \: }} \\ \\ \\ [/tex]
[tex]\large\underline{\sf{Solution-2}}[/tex]
[tex]\sf \: (2m-5n)^{2} - 10m + 25n \\ \\ [/tex]
[tex]\qquad\sf \: = \: (2m-5n)^{2} -5(2m - 5n) \\ \\ [/tex]
[tex]\qquad\sf \: = \: (2m-5n)(2m - 5n - 5) \\ \\ [/tex]
[tex]\boxed{ \sf{ \: (2m-5n)^{2} - 10m + 25n = (2m-5n)(2m - 5n - 5) \: }} \\ \\ \\ [/tex]
[tex]\large\underline{\sf{Solution-3}}[/tex]
[tex]\sf \: 3(x-2y)^{2} + 6(2y-x) \\ \\ [/tex]
[tex]\qquad\sf \: = 3(x-2y)^{2} - 6(x - 2y) \\ \\ [/tex]
[tex]\qquad\sf \: = 3(x-2y)^{2} - 3.2.(x - 2y) \\ \\ [/tex]
[tex]\qquad\sf \: = 3(x-2y) \: (x - 2y - 2) \\ \\ [/tex]
[tex]\boxed{ \sf{ \:3(x-2y)^{2} + 6(2y-x) = 3(x-2y) \: (x - 2y - 2) \: }} \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]
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Verified answer
Answer:
[tex]\qquad \:\boxed{\begin{aligned}& \qquad \:\sf \:(1) \: \: 3{(2a - 3b)} \: ( 8a - 12b- 5)\qquad \: \\ \\& \qquad \:\sf \:(2) \: \: (2m-5n)(2m - 5n - 5) \\ \\& \qquad \:\sf \:(3) \: \: 3(x-2y) \: (x - 2y - 2)\end{aligned}} \qquad \\ \\ [/tex]
Step-by-step explanation:
[tex]\large\underline{\sf{Solution-1}}[/tex]
Given algebraic expression is
[tex]\sf \: 12 {(2a - 3b)}^{2} - 15(2a - 3b) \\ \\ [/tex]
[tex]\qquad\sf \: = \: 2.2.3. {(2a - 3b)}^{2} - 3.5.(2a - 3b) \\ \\ [/tex]
[tex]\qquad\sf \: = \: 3{(2a - 3b)}[ 4(2a - 3b)- 5] \\ \\ [/tex]
[tex]\qquad\sf \: = \: 3{(2a - 3b)} \: ( 8a - 12b- 5) \\ \\ [/tex]
Hence,
[tex]\boxed{ \sf{ \:12 {(2a - 3b)}^{2} - 15(2a - 3b) = 3{(2a - 3b)} \: ( 8a - 12b- 5) \: }} \\ \\ \\ [/tex]
[tex]\large\underline{\sf{Solution-2}}[/tex]
Given algebraic expression is
[tex]\sf \: (2m-5n)^{2} - 10m + 25n \\ \\ [/tex]
[tex]\qquad\sf \: = \: (2m-5n)^{2} -5(2m - 5n) \\ \\ [/tex]
[tex]\qquad\sf \: = \: (2m-5n)(2m - 5n - 5) \\ \\ [/tex]
Hence,
[tex]\boxed{ \sf{ \: (2m-5n)^{2} - 10m + 25n = (2m-5n)(2m - 5n - 5) \: }} \\ \\ \\ [/tex]
[tex]\large\underline{\sf{Solution-3}}[/tex]
Given algebraic expression is
[tex]\sf \: 3(x-2y)^{2} + 6(2y-x) \\ \\ [/tex]
[tex]\qquad\sf \: = 3(x-2y)^{2} - 6(x - 2y) \\ \\ [/tex]
[tex]\qquad\sf \: = 3(x-2y)^{2} - 3.2.(x - 2y) \\ \\ [/tex]
[tex]\qquad\sf \: = 3(x-2y) \: (x - 2y - 2) \\ \\ [/tex]
Hence,
[tex]\boxed{ \sf{ \:3(x-2y)^{2} + 6(2y-x) = 3(x-2y) \: (x - 2y - 2) \: }} \\ \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]