# What is the standard form of y= (x +6)(x + 5)^2(x + 10)^2 ?

Dec 2, 2017

$y = {x}^{5} + 36 {x}^{4} + 505 {x}^{3} + 3450 {x}^{2} + 11500 x + 15000$

#### Explanation:

$y = \left(x + 6\right) {\left(x + 5\right)}^{2} {\left(x + 10\right)}^{2}$

FOIL ${\left(x + 5\right)}^{2}$:

$y = \left(x + 6\right) \left({x}^{2} + 10 x + 25\right) {\left(x + 10\right)}^{2}$

FOIL ${\left(x + 10\right)}^{2}$:

$y = \left(x + 6\right) \left({x}^{2} + 10 x + 25\right) \left({x}^{2} + 20 x + 100\right)$

Distribute the first two sections within parentheses:

$y = \left[\left(x + 6\right) \left({x}^{2}\right) + \left(x + 6\right) \left(10 x\right) + \left(x + 6\right) \left(25\right)\right] \left[{x}^{2} + 20 x + 100\right]$

Simplify:

$y = \left\{\left[\left({x}^{2}\right) \left(x\right) + \left({x}^{2}\right) \left(6\right)\right] + \left[\left(10 x\right) \left(x\right) + \left(10 x\right) \left(6\right)\right] + \left[\left(25\right) \left(x\right) + \left(25\right) \left(6\right)\right]\right\} \left[{x}^{2} + 20 x + 100\right]$

Simplify further:

$y = \left({x}^{3} + 6 {x}^{2} + 10 {x}^{2} + 60 x + 25 x + 150\right) \left({x}^{2} + 20 x + 100\right)$

Combine like terms within the first parentheses:

$y = \left({x}^{3} + 16 {x}^{2} + 85 x + 150\right) \left({x}^{2} + 20 x + 100\right)$

Distribute:

$y = \left[\left({x}^{2} + 20 x + 100\right) \left({x}^{3}\right)\right] + \left[\left({x}^{2} + 20 x + 100\right) \left(16 {x}^{2}\right)\right] + \left[\left({x}^{2} + 20 x + 100\right) \left(85 x\right)\right] + \left[\left({x}^{2} + 20 x + 100\right) \left(150\right)\right]$

Distribute further:

y={[(x^3)(x^2)]+[(x^3)(20x)]+[(x^3)(100)]}+{[(16x^2)(x^2)]]+[(16x^2)(20x)]+[(16x^2)(100)]}+{[(85x)(x^2)]+[(85x)(20x)]+[(85x)(100)]}+{[(150)(x^2)]+[(150)(20x)]+[(150)(100)]}

Simplify within brackets:

$y = {x}^{5} + 20 {x}^{4} + 100 {x}^{3} + 16 {x}^{4} + 320 {x}^{3} + 1600 {x}^{2} + 85 {x}^{3} + 1700 {x}^{2} + 8500 x + 150 {x}^{2} + 3000 x + 15000$

Combine like terms:

$y = {x}^{5} + 36 {x}^{4} + 505 {x}^{3} + 3450 {x}^{2} + 11500 x + 15000$