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Using binomial theorem, write down the e...

Using binomial theorem, write down the expansions of the following: `\ (1+2x-3x^2)^5`

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Given expansion term is `(1+2 x-3 x^2)^5`

This can be also written as `((1+2 x)-3 x^2)^5`

We know that, `(x+a)^n=sum_(r=0)^n n_(c_r) x^(n-r )a^r`

`=>5_(C_0)(1+2 x)^5(-3 x^2)^0+ 5_(C_1)(1+2 x)^4(-3 x^2)^1+5_(C_2)(1+2 x)^3(-3 x^2)^2+`

`5_(c_3)(1+2 x)^2(-3 x^2)^3+5_(c_4)(1+2 x)^1(-3 x^2)^4+5_(c_5)(1+2 x)^0(-3 x^2)^5`

`=>[5_(c_0)+5 c_1(2 x)^1+5_(c_2)(2 x)^2+5_(c_3)(2 x)^3+5_(c_4)(2 x)^4+5_(c_5)(2 x)^3]+`

`5[4_(c_0)(1)+4_(c_1)(2 x)+4_(c_2)(2 x)^2+4_(c_3)(2 x)^3+4_(c_4)(2 x)^4](-3 x^2)`

`+10[1+8 x^3+3 \cdot 2 x+3 \cdot 4 x^2)(9 x^-4)+10(1+4 x^2+4 x)(-27 x^6)`

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