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

Apr 6, 2018

$2 {a}^{3} + 8 {a}^{2} + 6 a$

#### Explanation:

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

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 $u = \left(3 + a\right)$, that way we have less to keep track of. Then we have:

$\left(2 a + 2 {a}^{2}\right) \left(3 + a\right) = \left(2 a + 2 {a}^{2}\right) \left(u\right) = u \left(2 a + 2 {a}^{2}\right)$.

Now we can use the familiar distributive property:

$u \left(2 a + 2 {a}^{2}\right) = u \left(2 a\right) + u \left(2 {a}^{2}\right) = 2 a u + 2 {a}^{2} u$.

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:

$2 a u + 2 {a}^{2} u = 2 a \left(3 + a\right) + 2 {a}^{2} \left(3 + a\right)$.

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

$2 a \left(3 + a\right) + 2 {a}^{2} \left(3 + a\right) = \left(6 a + 2 {a}^{2}\right) + \left(6 {a}^{2} + 2 {a}^{3}\right)$,
$= 2 {a}^{3} + 6 {a}^{2} + 2 {a}^{2} + 6 a = 2 {a}^{3} + 8 {a}^{2} + 6 a$.

Thus, our final answer is $2 {a}^{3} + 8 {a}^{2} + 6 a$.