(a) By the definition of conjugation, \[(u+w)^*=((a+c)+i(b+d))^*=(a+c)-i(b+d).\] Also \[u^*+w^*=(a-ib)+(c-id)=(a+c)-i(b+d).\] The two expressions are equal, proving the result.

(b) Put z=x+iy. Then \[(z+2+i)^*=x+2-i(y+1),\] \[(2+i)z=(2x-y)+i(x+2y).\] Equating real and imaginary parts to zero gives \[3x-y+2=0,\qquad x+y-1=0.\] Hence y=1−x and 4x+1=0, so \[\boxed{z=-\frac14+\frac54i}.\]