Question

How do you find the product of `(2h+3)(2h^2+3h+4)`?

Answer 1

`4h^3+12h^2+17h+12`

Explanation:

Each term in the 2nd bracket must be multiplied by each term in the 1st bracket.

The following shows how this may be done.

`(color(red)(2h+3))(2h^2+3h+4)`

`=color(red)(2h)(2h^2+3h+4)color(red)(+3)(2h^2+3h+4)`

now distribute each pair of brackets.

`=4h^3+6h^2+8h+6h^2+9h+12`

and collecting like terms gives.

`=4h^3+(6h^2+6h^2)+(8h+9h)+12`

`=4h^3+12h^2+17h+12larr"result of product"`

Answer 2

`(2h+3)(2h^2+3h+4)=color(orange)(4h^3+12h^2+17h+12)`

Explanation:

`color(red)(""(2h+3))(2h^2+3h+4)`

using the distributive property
`color(white)("XXX")=color(red)(2h)(2h^2+3h+4)color(red)(+3)(2h^2+3h+4)`

(no real change except color to see the separate terms more clearly)
`color(white)("XXX")=color(blue)(2h(2h^2+3h+4))+color(green)(3(2h^2+3h+4)`

expanding each term
`color(white)("XXX")=color(blue)(4h^3+6h^2+8h)+color(green)(6h^2+9h+12)`

grouping equal powers of the variable
`color(white)("XXX")=color(blue)(4h^3)+(color(blue)(6h^2)+color(green)(6h^2))+(color(blue)(8h)+color(green)(9h))+(color(green)(12))`

Adding together equal powers of the variable
`color(white)("XXX")=4h^3+12h^2+17h+12`