Question

How do you find the product `(2n^2+3n-6)(5n^2-2n-8)`?

Answer 1

Multiply each of the terms inside the left parenthesis by each of the terms inside the right parenthesis, group and then combine like terms. See full process below:

Explanation:

To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis.

`(color(red)(2n^2) + color(red)(3n) - color(red)(6))(color(blue)(5n^2) - color(blue)(2n) - color(blue)(8))` becomes:

`(color(red)(2n^2) xx color(blue)(5n^2)) - (color(red)(2n^2) xx color(blue)(2n)) - (color(red)(2n^2) xx color(blue)(8)) + (color(red)(3n) xx color(blue)(5n^2)) - (color(red)(3n) xx color(blue)(2n)) - (color(red)(3n) xx color(blue)(8)) - (color(red)(6) xx color(blue)(5n^2)) + (color(red)(6) xx color(blue)(2n)) + (color(red)(6) xx color(blue)(8))`

`10n^4 - 4n^3 - 16n^2 + 15n^3 - 6n^2 - 24n - 30n^2 + 12n + 48`

Next, We can group like terms:

`10n^4 - 4n^3 + 15n^3 - 16n^2 - 6n^2 - 30n^2 - 24n + 12n + 48`

Now, we can combine like terms:

`10n^4 + (-4 + 15)n^3 + (-16 - 6 - 30)n^2 + (-24 + 12)n + 48`

`10n^4 + 11n^3 - 52n^2 - 12n + 48`

Answer 2

`10n^4+11n^3-52n^2-12n+48`

Explanation:

Using tabular multiplication:

`color(white)("XX")underline(color(white)("X")xxcolor(white)("X")|color(white)("XX")2n^2color(white)("X")+3ncolor(white)("XX)-6)`
`color(white)("XXX")5n^2color(white)("X")|color(white)("X")color(red)(10n^2)color(white)("X")color(blue)(+15n^3)color(white)("X")color(green)(-30n^2)`
`color(white)("XX")-2ncolor(white)("X")|color(white)("X")color(blue)(-4n^3)color(white)("X")color(green)(-6n^2)color(white)("X")color(magenta)(+12n)`
`color(white)("Xx")underline(color(white)("X")-8color(white)("X")|color(white)("X")color(green)(-16n^2)color(white)("X")color(magenta)(-24n)color(white)("X")color(orange)(+48))`
`color(red)(10n^4)color(white)("X")color(blue)(+11n^3)color(white)("X")color(green)(-52n^2)color(white)("X")color(magenta)(-12n)color(white)("X")color(orange)(+48)`