1. Find multiplication of `10x+25` and `x-5`

Solution:
Method-1: Multiply using the Distributive property
`(10x+25)*(x-5)`

`=10x*(x-5)+25*(x-5)` [Distribute `(x-5)`]

`=10x^2-50x+25x-125` (Distribute again)

`=color{blue}{ 10x^2}color{green}{ - 50x}color{green}{ + 25x}color{red}{ - 125}` (Identify like terms)

`=color{blue}{ 10x^2}color{green}{ - 50x}color{green}{ + 25x}color{red}{ - 125}` (Rearrange terms to get like terms together)

`=color{blue}{ 10x^2}color{green}{ - 25x}color{red}{ - 125}` (Combine like terms)

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Method-2: Multiply using the Vertical method

Solution using Vertical method

	`x-5`
`xx`	`10x+25`
	`25x-125`	`larr` Multiply `(x-5)` by `25`
	`10x^2-50x`	`larr` Multiply `(x-5)` by `10x`
	`10x^2-25x-125`	`larr` Add like terms

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Method-3: Multiply using the FOIL method

Solution using FOIL method : `(a+b)(c+d)=ac+ad+bc+bd`

`(10x+25)(x-5)`

`((10x+25)(x-5))`

FOIL method : `(a+b)(c+d)=ac+ad+bc+bd`

Multiply First, Outer, Inner, Last :
(F) First, multiply first terms of each binomial: `color{red}{10x}*color{green}{x}=10x^2`

(O) Outside, multiply outside terms of the binomial: `color{red}{10x}*color{purple}{(-5)}=(-50x)`

(I) Inside, multiply inside terms of the binomial: `color{blue}{25}*color{green}{x}=25x`

(L) Last, multiply last terms of each binomial: `color{blue}{25}*color{purple}{(-5)}=(-125)`

Put the terms together
`=10x^2+(-50x)+25x+(-125)`

Combine like terms to simpplify
`=10x^2-25x-125`

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Solution Method-2 of FOIL method

FOIL method : `(a+b)(c+d)=ac+ad+bc+bd`

`=(color{red}{10x}+color{blue}{25})(color{green}{x}+color{purple}{(-5)})`

`=color{red}{10x}*(color{green}{x}+color{purple}{(-5)})+color{blue}{25}*(color{green}{x}+color{purple}{(-5)})`

`=color{red}{10x}*color{green}{x}+color{red}{10x}*color{purple}{(-5)}+color{blue}{25}*color{green}{x}+color{blue}{25}*color{purple}{(-5)}`

`=10x^2+(-50x)+25x+(-125)`

`=10x^2-25x-125`

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2. Find multiplication of `x-2` and `x^2+4x-8`

Solution:
Method-1: Multiply using the Distributive property
`(x-2)*(x^2+4x-8)`

`=x*(x^2+4x-8)-2*(x^2+4x-8)` [Distribute `(x^2+4x-8)`]

`=x^3+4x^2-8x-2x^2-8x+16` (Distribute again)

`=color{maroon}{ x^3}color{blue}{ + 4x^2}color{green}{ - 8x}color{blue}{ - 2x^2}color{green}{ - 8x}color{red}{ + 16}` (Identify like terms)

`=color{maroon}{ x^3}color{blue}{ + 4x^2}color{blue}{ - 2x^2}color{green}{ - 8x}color{green}{ - 8x}color{red}{ + 16}` (Rearrange terms to get like terms together)

`=color{maroon}{ x^3}color{blue}{ + 2x^2}color{green}{ - 16x}color{red}{ + 16}` (Combine like terms)

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Method-2: Multiply using the Vertical method

Solution using Vertical method

	`x^2+4x-8`
`xx`	`x-2`
	`-2x^2-8x+16`	`larr` Multiply `(x^2+4x-8)` by `-2`
	`x^3+4x^2-8x`	`larr` Multiply `(x^2+4x-8)` by `x`
	`x^3+2x^2-16x+16`	`larr` Add like terms

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3. Find multiplication of `x^2-4x-16` and `x^2+x-8`

Solution:
Method-1: Multiply using the Distributive property
`(x^2-4x-16)*(x^2+x-8)`

`=x^2*(x^2+x-8)-4x*(x^2+x-8)-16*(x^2+x-8)` [Distribute `(x^2+x-8)`]

`=x^4+x^3-8x^2-4x^3-4x^2+32x-16x^2-16x+128` (Distribute again)

`=color{fuchsia}{ x^4}color{maroon}{ + x^3}color{blue}{ - 8x^2}color{maroon}{ - 4x^3}color{blue}{ - 4x^2}color{green}{ + 32x}color{blue}{ - 16x^2}color{green}{ - 16x}color{red}{ + 128}` (Identify like terms)

`=color{fuchsia}{ x^4}color{maroon}{ + x^3}color{maroon}{ - 4x^3}color{blue}{ - 8x^2}color{blue}{ - 4x^2}color{blue}{ - 16x^2}color{green}{ + 32x}color{green}{ - 16x}color{red}{ + 128}` (Rearrange terms to get like terms together)

`=color{fuchsia}{ x^4}color{maroon}{ - 3x^3}color{blue}{ - 28x^2}color{green}{ + 16x}color{red}{ + 128}` (Combine like terms)

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Method-2: Multiply using the Vertical method

Solution using Vertical method

	`x^2+x-8`
`xx`	`x^2-4x-16`
	`-16x^2-16x+128`	`larr` Multiply `(x^2+x-8)` by `-16`
	`-4x^3-4x^2+32x`	`larr` Multiply `(x^2+x-8)` by `-4x`
	`x^4+x^3-8x^2`	`larr` Multiply `(x^2+x-8)` by `x^2`
	`x^4-3x^3-28x^2+16x+128`	`larr` Add like terms

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4. Find multiplication of `x^2+2x+3` and `x^2+x-1`

Solution:
Method-1: Multiply using the Distributive property
`(x^2+2x+3)*(x^2+x-1)`

`=x^2*(x^2+x-1)+2x*(x^2+x-1)+3*(x^2+x-1)` [Distribute `(x^2+x-1)`]

`=x^4+x^3-x^2+2x^3+2x^2-2x+3x^2+3x-3` (Distribute again)

`=color{fuchsia}{ x^4}color{maroon}{ + x^3}color{blue}{ - x^2}color{maroon}{ + 2x^3}color{blue}{ + 2x^2}color{green}{ - 2x}color{blue}{ + 3x^2}color{green}{ + 3x}color{red}{ - 3}` (Identify like terms)

`=color{fuchsia}{ x^4}color{maroon}{ + x^3}color{maroon}{ + 2x^3}color{blue}{ - x^2}color{blue}{ + 2x^2}color{blue}{ + 3x^2}color{green}{ - 2x}color{green}{ + 3x}color{red}{ - 3}` (Rearrange terms to get like terms together)

`=color{fuchsia}{ x^4}color{maroon}{ + 3x^3}color{blue}{ + 4x^2}color{green}{ + x}color{red}{ - 3}` (Combine like terms)

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Method-2: Multiply using the Vertical method

Solution using Vertical method

	`x^2+x-1`
`xx`	`x^2+2x+3`
	`3x^2+3x-3`	`larr` Multiply `(x^2+x-1)` by `3`
	`2x^3+2x^2-2x`	`larr` Multiply `(x^2+x-1)` by `2x`
	`x^4+x^3-x^2`	`larr` Multiply `(x^2+x-1)` by `x^2`
	`x^4+3x^3+4x^2+x-3`	`larr` Add like terms

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