Multiplying out $(x^3 + 3x^2 + 3x + 1) \times (1 + x + x^2)$ I know this is a basic math question but still I always get stuck on this.
$(x^3 + 3x^2 + 3x + 1) \times (1 + x + x^2)$ 
I know this is the solution:
$1 + 4x + 7x^2 + 7x^3 + 4x^4 + x^5$
But I want to know how they got there.
This has to be solved with distributing the second term over the first one (I guess).
So I get these terms:
$x^3 + 3x^2 + 3x + 1$
$x^4 + 3x^3 + 3x^2 + 1x$
$x^5 + 3x^4 + 3x^3 + 1x^2$
And than I'd have to sum these up.
I already highly doubt this is correct, but even if it was correct I wouldn't know how to sum these up, because it's not possible to sum numbers with a different exponent.
Could someone explain step-by-step on how to solve this $(x^3 + 3x^2 + 3x + 1) \times (1 + x + x^2)$ 

Yes, you are right, so far. But, why can’t you just sum up terms with same power of $x$, that is, $$(x^3+3x^2+3x+1)(1+x+x^2) $$ $$= (x^\color{red}{5})+(3x^\color{green}{4} + x^\color{green}{4}) +  (x^\color{blue}{3}+3x^\color{blue}{3}+3x^\color{blue}{3})+(3x^\color{orange}{2}+3x^\color{orange}{2}+x^\color{orange}{2})+(3x^\color{cyan}{1}+x^\color{cyan}{1})+1 $$ $$= x^\color{red}{5}+4x^\color{green}{4}+7x^\color{blue}{3}+7x^\color{orange}{2}+4x^\color{cyan}{1}+1$$
