Question: Factorize the expression:

[tex]\sf \: (3a - 5b)^{2} + 2(3a - 5b)(2b - a) + (2b - a)^{2} \\ \\ [/tex]

Answer:

[tex]\boxed{\sf \: \: {(2a - 3b)}(2a - 3b) \: } \\ \\ [/tex]

Step-by-step explanation:

Given algebraic expression is

[tex]\sf \: (3a - 5b)^{2} + 2(3a - 5b)(2b - a) + (2b - a)^{2} \\ \\ [/tex]

Let assume that

[tex]\sf \: x = 3a - 5b \\ [/tex]

and

[tex]\sf \: y = 2b - a \\ [/tex]

So, above expression can be rewritten as

[tex]\sf \: = \: {x}^{2} + 2xy + {y}^{2} \\ \\ [/tex]

[tex]\sf \: = \: {(x + y)}^{2} \\ \\ [/tex]

[tex]\sf \: = \: {(3a - 5b + 2b - a)}^{2} \\ \\ [/tex]

[tex]\sf \: = \: {(2a - 3b)}^{2} \\ \\ [/tex]

[tex]\sf \: = \: {(2a - 3b)}(2a - 3b) \\ \\ [/tex]

Hence,

[tex]\boxed{\sf \: (3a - 5b)^{2} + 2(3a - 5b)(2b - a) + (2b - a)^{2} = \: {(2a - 3b)}(2a - 3b) \: } \\ \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Additional Information

[tex]{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{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} }} \\ \\ [/tex]