Expand each of the following,using suitable identities: $\left[\frac{1}{4} a-\frac{1}{2} b+1\right]^{2}$

We use the algebraic identity $(x+y+z)^{2} = x^{2}+y^{2}+z^{2}+2xy+2yz+2zx$.
Here,$x = \frac{1}{4}a$,$y = -\frac{1}{2}b$,and $z = 1$.
Substituting these values into the identity:
$\left[\frac{1}{4} a-\frac{1}{2} b+1\right]^{2} = \left(\frac{1}{4} a\right)^{2} + \left(-\frac{1}{2} b\right)^{2} + (1)^{2} + 2\left(\frac{1}{4} a\right)\left(-\frac{1}{2} b\right) + 2\left(-\frac{1}{2} b\right)(1) + 2(1)\left(\frac{1}{4} a\right)$
$= \frac{1}{16} a^{2} + \frac{1}{4} b^{2} + 1 - \frac{1}{4} ab - b + \frac{1}{2} a$.