Question

Question #15c3f

Answer 1

`x^3+3x^2-4`

Explanation:

`(x-1)(x+2)(x+2)`

`(x+2)^2(x-1)`

Repeated expansion now helps.

`=(x^2+4x+4)(x-1)`

`=x^3-x^2+4x^2-4x+4x-4`

`=x^3+3x^2-cancel(4x)+cancel(4x)-4`

`=x^3+3x^2-4`

Answer 2

`x^3+3x^2-4`

Explanation:

`color(blue)((x-1))color(green)( (x+2) )color(red)((x+2)`

Consider just the first two pairs of brackets.

Multiply everything inside the green brackets by everything in the blue.

`color(green)(color(blue)(x)(x+2)color(white)("ddd") color(blue)(-1)(x+2))`

Notice the way the minus sign followed the 1

`x^2+2xcolor(white)("ddd")-x-2`

`x^2+x-2color(white)("d")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So putting it all back together we have:

`color(purple)((x^2+x-2)) color(red)( (x+2))`

Multiply everything in the red brackets by everything in the purple.

`color(red)(color(purple)(x^2)(x+2)color(white)("ddd")color(purple)(+x)(x+2)color(white)("ddd")color(purple)(-2)(x+2) )`

`x^3+2x^2color(white)("dddd")+x^2+cancel(2x)color(white)("dd")cancel(-2x)-4`

`x^3+3x^2-4`

Answer 3

`x^3+3x^2-4`

Explanation:

`"the expansion of factors in the form "`

`(x+a)(x+b)(x+c)`

`=x^3+(a+b+c)x^2+(ab+bc+ac)x+abc`

`"here "a=-1,b=c=2`

`rArr(x-1)(x+2)(x+2)`

`=x^3+(-1+2+2)x^2+(-2+4-2)x+( -1.2.2)`

`=x^3+3x^2-4`