Answer:
[tex]\boxed{ \sf{ \:( \sqrt{7} + \sqrt{3})( \sqrt{5} - \sqrt{12}) = \sqrt{35} - 2\sqrt{21} + \sqrt{15} - 6 \: }}\\ \\ [/tex]
Step-by-step explanation:
Given expression is
[tex]\sf \:( \sqrt{7} + \sqrt{3})( \sqrt{5} - \sqrt{12}) \\ \\ [/tex]
[tex]\sf \: = \: \sqrt{7}( \sqrt{5} - \sqrt{12}) + \sqrt{3}( \sqrt{5} - \sqrt{12})\\ \\ [/tex]
[tex]\sf \: = \: \sqrt{35} - \sqrt{84} + \sqrt{15} - \sqrt{36} \\ \\ [/tex]
[tex]\sf \: = \: \sqrt{35} - \sqrt{2 \times 2 \times 3 \times 7} + \sqrt{15} - \sqrt{6 \times 6} \\ \\ [/tex]
[tex]\sf \: = \: \sqrt{35} - 2\sqrt{21} + \sqrt{15} - 6 \\ \\ [/tex]
Hence,
[tex]\boxed{ \sf{ \:( \sqrt{7} + \sqrt{3})( \sqrt{5} - \sqrt{12}) = \sqrt{35} - 2\sqrt{21} + \sqrt{15} - 6 \: }}\\ \\ [/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]