Question

How do you multiply `(2a + 2a^2)(3 + a)`?

Answer 1

`2a^3 + 8a^2 + 6a`

Explanation:

To multiply this polynomial, you must use the distributive property. Recall that a polynomial like `4(x + 2) = 4(x) + 4(2) = 4x + 8`.

To use the distributive property in a polynomial like the one you gave, it helps to "simplify" it to an easier form. Let's let `u = (3 + a)`, that way we have less to keep track of. Then we have:

`(2a + 2a^2)(3+a) = (2a + 2a^2)(u) = u(2a + 2a^2)`.

Now we can use the familiar distributive property:

`u(2a + 2a^2) = u(2a) + u(2a^2) = 2au + 2a^2u`.

We now have `u` in our answer, which we don't want. Remember that we let `u = 3 + a`, so we can replace every `u` with a `3 + a`.

This gives:

`2au + 2a^2u = 2a(3+a) + 2a^2(3+a)`.

We can see that we now have to use the distributive property again, twice this time. This gives:

`2a(3+a) + 2a^2(3+a) = (6a + 2a^2) + (6a^2 + 2a^3)`,
`= 2a^3 + 6a^2 + 2a^2 + 6a = 2a^3 + 8a^2 + 6a`.

Thus, our final answer is `2a^3 + 8a^2 + 6a`.