Hard Indices question

Im going through exercises in the A level MEI maths Year 1 book and Im stuck on this particular question in the topic "Working with indices".
(1+x)^1/2 + (1+x)^3/2 simplify.

I got to the second step where you use the exponent rule a^b+c = (a)^b(a)^c
to simplify the whole expression into:
(1+x)^1/2 + (1+x)(1+x)^1/2
I googled this and the next step is to "factor out the common term (1+x)^1/2 to arrive to an answer of (x+2)(x+1)^1/2 and I have no idea how they got to this answer, would appreciate if someone explained the jump and what it means

next step:
$(1+x)^{1/2}[1+(1+x)]$ -You are taking out a factor of (1+x)^1/2 (if you dont understand this, multipy the square bracket by the (1+x) and you will understand).

$(1+x)^{1/2}(2+x)$ you add up what is in the square bracket

:/ I still don't understand what you mean by taking out a factor of (1+x)^1/2

When you have
$5x^3+10x^2+5x$ - you can take out a factor of $5x$
That will leave you with : $5x(x^2+2x+1)$ you are just removing(dividing) the whole equation by 5x
This is the same thing that they are doing with your Equation.

if you expand this you get your original equation of $5x^3+10x^2+5x$

If you still dont understand watch this: https://www.youtube.com/watch?v=iOHYuBoWwTY skip to 12:06 and watch him factor out $5(x^3-1)^3 from....60x^3(x^3-1)^3+5(x^3-1)^4$

With your equation
$(1+x)^{1/2}+(1+x)^{3/2}$ just factor out (1+x)^1/2
$(1+x)^{1/2}[1+(1+x)^1]$ simplify the square bracket
$(1+x)^{1/2}(x+2)$
i hope someone else replies and helps you, if you still dont understand.