Question

How do you multiply `(2a ^ { 2} + 4a + 1) ( a ^ { 2} - 6a + 5)`?

Answer 1

`=2a^4-8a^3-13a^2+14a+5`

Explanation:

Each term in the first bracket has to be multiplied by each term in the second. This is using the distributive law.

You will end up with `3xx3=9` terms

`(color(red)(2a^2)color(blue)(+4a)color(green)(+1))(a^2-6a+5)`

`=color(red)(2a^2)(a^2-6a+5)color(blue)(+4a)(a^2-6a+5)color(green)(+1)(a^2-6a+5)`

`=color(red)(2a^4-12a^3+10a^2)color(blue)(+4a^3-24a^2+20a)color(green)(+a^2-6a+5)`

Now simplify by adding all the like terms.

`=2a^4-8a^3-13a^2+14a+5`

Answer 2

`2a^4-8a^3-13a^2+14a+5`

Explanation:

Expanding the product of two trinomials like these involves the repeated application of the distributive property which, as you might remember, states in its simplest form that, given three real numbers `a`, `b`, and `c`, `a(b+c)=ab+ac`. This extends beyond two-number groupings, though. We could have just as easily selected a fourth real number, `d`, and `a(b+c+d)=ab+ac+ad` would be an equally valid statement.

Here, we can treat the trinomial `2a^2+4a+1` as the `a` term, and the terms `a^2`, `-6a`, and `5` from the second trinomial as the `b`, `c`, and `d` terms. When we distribute that first trinomial across those three terms, we get:

`(2a^2+4a+1)a^2+(2a^2+4a+1)(-6a)+(2a^2+4a+1)5`

Here, we can distribute again to obtain

`2a^2(a^2)+4a(a^2)+a^2+2a^2(-6a)+4a(-6a)-6a+2a^2(5)+4a(5)+5=`
`=2a^4+4a^3+a^2-12a^3-24a^2-6a+10a^2+20a+5`

Collecting and grouping like terms:

`color(indigo)(2a^4)color(tomato)(+4a^3)color(dodgerblue)(+a^2)color(tomato)(-12a^3)color(dodgerblue)(-24a^2)color(forestgreen)(-6a)color(dodgerblue)(+10a^2)color(forestgreen)(+20a)color(sandybrown)(+5)`

`color(indigo)(2a^4)color(tomato)(+4a^3)color(tomato)(-12a^3)color(dodgerblue)(+a^2)color(dodgerblue)(-24a^2)color(dodgerblue)(+10a^2)color(forestgreen)(-6a)color(forestgreen)color(forestgreen)(+20a)color(sandybrown)(+5)`

And finally, simplifying:

`2a^4-8a^3-13a^2+14a+5`