$(x+1{)}^2$ :

Design a combinational circuit that computes $P=(x+1{)}^2=x^2+2x+1$

The input $x$ is a $4-$bit unsigned integer. The output $P$ is an $8-$bit unsigned integer.

Your design should use

a carry$-$save tree using half and full adders and

a ripple adder using half and full adders.

a$)$ Show how partial sums are formed from the input bits, $x_i.$

Simplify using the following identities. abo$+a=\frac{a}{b}ar(b)$

$ab*a=ab$

$ao+1=$

$a*1=a$

use the notation $($ij$)$denote $x_i*x_j$ where $i,$jin$\{0,1,2,3\}$

for example $(3\frac{?}{\text{b}\text{a}\text{r}}(2))$ denotes $x_3\frac{*}{b}ar(x_2)$

b$)$ Connect the adders. You do not need to draw the AND gates, simply label the nodes using the

notation given above. Label the output bits, $P_i.$