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Here is your Question in the above attachment.

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Answer 1

2.

$\sf {\mapsto \frac{2a}{ {a}^{2} - 4 {x}^{2} } + \frac{1}{2 {x}^{2} + 6x - ax - 3a} \left( x + \frac{3x - 6}{x - 2} \right)}$

$\sf{ \mapsto \frac{2a}{ {a}^{2} - 4 {x}^{2} } + \frac{1}{2 {x}^{2} + 6x - ax - 3a } \left( x + \frac{3 \cancel{(x - 2)}}{ \cancel{x - 2}} \right) }$

$\sf{ \mapsto \frac{2a}{ {a}^{2} - 4 {x}^{2} } + \frac{1}{2 {x}^{2} + 6x - ax - 3a } \left( x + 3\right) }$

$\sf{ \mapsto \frac{2a}{ {a}^{2} - 4 {x}^{2} } + \frac{x + 3}{2 {x}^{2} + 6x - ax - 3a } }$

$\sf \footnotesize \green{ Multiply \: \frac{x + 3}{( - x - 3)( - 2x + a)} \: \: by \: \frac{2x + a}{2x + a} }$

$\sf{ \mapsto \frac{2a( - x - 3)}{( - x - 3)( - 2x + a)(2x + a)} + \frac{(x + 3)(2x + a)}{( - x - 3)( - 2x + a)(2x + a)} }$

$\sf{ \mapsto \frac{2a( - x - 3) + (x + 3)(2x + a)}{( - x - 3)( - 2x + a)(2x + a)} }$

$\sf{ \mapsto \frac{- xa - 3a + 6x + 2 {x}^{2}}{( - x - 3)( - 2x - a)(2x + a)} }$

$\sf{ \mapsto \frac{ \cancel{( - x - 3)( - 2x + a)}}{ \cancel{( - x - 3)( - 2x + a)}(2x + a)} }$

$\sf \large \pink{ \mapsto \frac{1}{2x + a} }$

3.

$\sf{ \mapsto \left( \frac{2a + 10}{3a - 1} + \frac{130 - a}{1 - 3a} + \frac{30}{a} - 3 \right) \times \frac{3 {a}^{3} + 8 {a}^{2} - 3a}{1 - \frac{1}{4} {a}^{2} } }$

$\sf{ \mapsto \frac{(3 {a}^{3} + 8 {a}^{2} - 3a ) \left( \frac{130 - a}{1 - 3a} + \frac{30}{a} + \frac{2a + 10}{3a - 1} - 3\right)}{1 - \frac{ {a}^{2} }{4} } }$

$\sf \footnotesize \green{After \: solving \: the \: numerator,}$

$\sf{ \mapsto \frac{ 3( - a - 2)(a + 3)(2a + 5)}{1 - \frac{1}{4} {a}^{2} } }$

$\sf{ \mapsto \frac{ 3 \cancel{( - a - 2)}(a + 3)(2a + 5)}{\frac{1}{4} (a - 2) \cancel{( - a - 2)} } }$

$\sf \large \pink{ \mapsto \frac{ 3(a + 3)(2a + 5)}{\frac{1}{4} (a - 2) } }$

4.

$\sf{ \mapsto \frac{ {a}^{2} - {b}^{2} }{a - b} - \frac{ {a}^{3} - {b}^{3} }{ {a}^{2} - {b}^{2} } }$

$\sf{ \mapsto \frac{ (a + b) \cancel{( a-b ) }}{ \cancel{a - b}} - \frac{ {a}^{3} - {b}^{3} }{ {a}^{2} - {b}^{2} } }$

$\sf{ \mapsto (a + b) - \frac{ {a}^{3} - {b}^{3} }{ {a}^{2} - {b}^{2} } }$

$\sf{ \mapsto (a + b) - \frac{ \cancel{( a-b )}( {a}^{2} +ab + {b}^{2} ) }{(a + b) \cancel{(a -b )}} }$

$\sf{ \mapsto a + b - \frac{ {a}^{2} + ab + {b}^{2} }{a + b} }$

$\sf{ \mapsto \frac{( a+ b)(a +b ) - ( {a}^{2} + ab+ {b}^{2} )}{ a+ b} }$

$\sf{ \mapsto \frac{( a+ b)^{2}- ( {a}^{2} + ab+ {b}^{2} )}{ a+ b} }$

$\sf{ \mapsto \frac{ {a}^{2} + ab + ab + {b}^{2} - {a}^{2} - ab - {b}^{2} }{ a+ b} }$

$\sf {\large {\pink{ \mapsto \frac{ab}{a + b} }}}$

$\hookrightarrow \textsf{Identities used in 4th part}$

$\sf{ \implies(a + b)^{2} = {a}^{2} + 2ab+ {b}^{2} } \\ \sf{ \implies(a - b)^{2} = {a}^{2} - 2ab+ {b}^{2} }$

$\sf{ \implies {a}^{3} - {b}^{3} = ( a- b)( {a}^{2} + ab+ {b}^{2} ) }$

$\sf{ \implies (a - b)(a + b) = {a}^{2} - {b}^{2} }$

$\hookrightarrow \textsf{Note :}$

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