Find the coefficient of `x^(4)` in the product `(1+2x)^(4)(2-x)^(5)` by using binomial theorem.

Explanation

We know that
`(1+2x)^(40+.^(4)C_(0)+.^(4)C_(1)(2x)+.^(4)C_(2)(2x)^(2)+.^(4)C_(3)(2x)^(3)+.^(4)C_(4)(2x)^(4))`
`=1+4(2x)+6(2x)^(2)+4(2x)^(3)+16x^(4)`
`=1+8x+24x^(2)+32x^(3)+16x^(4)`
and `(2-x)^(5)=.^(5)C_(0)(2)^(5)-.^(5)C_(1)(2)^(4)(x)+.^(5)C_(2)(2)^(3)(x^(2))-.^(5)C_(3)(2)^(2)(x^(3))+.^(5)C_(4)(2)(x^(4))-.^(5)C_(5)(x)^(5)`
`=32-80x+80x^(2)-40x^(3)+10x^(4)-x^(5)`
Thus, `(1+2x^(2))^(4)(2-x)^(5)={1+8x+24x^(2)+32x^(3)+16x^(4)}`
`{32-80x+80x^(2)-40x^(3)+10x^(4)-x^(5)}`
Now, we have to find the coefficient of `x^(4)`, so the complete multiplication of the above two need not to be done. we write only those terms which involve `x^(4)`. the terms containing `x^(4)` are
`1.(10x^(4))+(8x)(-40x^(3))+(24x^(2))(80x^(2))+(32x^(3))(-80x)+(16x^(4))(32)=-438x^(4)`
thus, the coefficient of `x^(4)` in the given product is -438.