How do I do this sort of factoring

like one of the examples is:
1/2(X+1) + X(X+1) factors into 1/2(X+1)[1+2x]

Can someone explain this more?

( x+1) ( 1/2 + x)

= ( x+1) ( (1 + 2x)/2)

= 1/2 ( x+1) ( 1 + 2x) ANSWER

½(x+1) + x(x+1) = (x+1)(x + ½)

Now if you want to get rid of the fraction, then multiply and divide by 2. Or in other words, multiply by 1 in the form ½•2, so

½•2(x +1)(x + ½) = ½(x +1)•2(x + ½) =

½(x +1)(2x + 1)

Or you could just add x + ½ as 2x/2 + 1/2 = (2x + 1)/2. Then the denominator of 2 comes out front as the ½.