[tex]\large\underline{\sf{Solution-}}[/tex]

Given expression is

[tex]\sf \: {(3x - 1)}^{2} + 12x \\ \\ [/tex]

We know,

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

So, using this identity, we get

[tex]\sf \: = \: {(3x)}^{2} + {(1)}^{2} - 2(3x)(1) + 12x \\ \\ [/tex]

[tex]\sf \: = \: {(3x)}^{2} + {(1)}^{2} -6x + 12x \\ \\ [/tex]

[tex]\sf \: = \: {(3x)}^{2} + {(1)}^{2}+ 6x \\ \\ [/tex]

[tex]\sf \: = \: {(3x)}^{2} + {(1)}^{2}+ 2(3x) \\ \\ [/tex]

[tex]\sf \: = \: {(3x)}^{2} + {(1)}^{2}+ 2(3x)(1) \\ \\ [/tex]

We know,

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

So, using this identity, we get

[tex]\sf \: = \: {(3x + 1)}^{2} \\ \\ [/tex]

Hence,

[tex]\sf \: \sf \:  \implies \: \boxed{ \sf{ \: {(3x - 1)}^{2} + 12x = {(3x + 1)}^{2} \: }}\\ \\ [/tex]

Alternative Method

Given expression is

[tex]\sf \: {(3x - 1)}^{2} + 12x \\ \\ [/tex]

can be rewritten as

[tex]\sf \: = \: {(3x - 1)}^{2} + 4 \times 3x \\ \\ [/tex]

[tex]\sf \: = \: {(3x - 1)}^{2} + 4 \times 3x \times 1 \\ \\ [/tex]

[tex]\sf \: = \: {(3x - 1)}^{2} + 4(3x)(1) \\ \\ [/tex]

We know,

[tex]\boxed{ \sf{ \: {(x + y)}^{2} = {(x - y)}^{2} + 4xy \: }} \\ \\ [/tex]

So, using this identity, we get

[tex]\sf \: = \: {(3x + 1)}^{2} \\ \\ [/tex]

Hence,

[tex]\sf \: \sf \:  \implies \: \boxed{ \sf{ \: {(3x - 1)}^{2} + 12x = {(3x + 1)}^{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]

Hi

The Question Say is :-

Express as a perfect square

(3x-1)² +12x

Answer :-

• Solution given in figure .