To multiply this polynomial, you must use the distributive property. Recall that a polynomial like 4(x + 2) = 4(x) + 4(2) = 4x + 84(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)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)(2a+2a2)(3+a)=(2a+2a2)(u)=u(2a+2a2).
Now we can use the familiar distributive property:
u(2a + 2a^2) = u(2a) + u(2a^2) = 2au + 2a^2uu(2a+2a2)=u(2a)+u(2a2)=2au+2a2u.
We now have uu in our answer, which we don't want. Remember that we let u = 3 + au=3+a, so we can replace every uu with a 3 + a3+a.
This gives:
2au + 2a^2u = 2a(3+a) + 2a^2(3+a)2au+2a2u=2a(3+a)+2a2(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+a)+2a2(3+a)=(6a+2a2)+(6a2+2a3),
= 2a^3 + 6a^2 + 2a^2 + 6a = 2a^3 + 8a^2 + 6a=2a3+6a2+2a2+6a=2a3+8a2+6a.
Thus, our final answer is 2a^3 + 8a^2 + 6a2a3+8a2+6a.
