The image contains a list of algebraic expressions to be simplified or expanded. Here are the expressions rewritten for clarity:

1. Simplify:
a. \((x + 2y)^2\)
b. \((2x + 3y)^2\)
c. \(\left(2x - \frac{1}{2}\right)^2\)
d. \(\left(x^2 + \frac{2}{5}y\right)\left(x^2 - \frac{2}{5}y\right)\)

2. Simplify:
a. \(\left(x + \frac{1}{3}\right)^3\)
b. \(\left(2x + \frac{1}{2}\right)^3\)
c. \((x - 3y)(x^2 + 3xy + 9y^2)\)
d. \(\left(\frac{1}{3}x + 2y\right)\left(\frac{1}{9}x^2 - \frac{2}{3}xy + 4y^2\right)\)

Let's simplify each expression:

1. Simplify:
a. \((x + 2y)^2 = x^2 + 4xy + 4y^2\)
b. \((2x + 3y)^2 = 4x^2 + 12xy + 9y^2\)
c. \(\left(2x - \frac{1}{2}\right)^2 = 4x^2 - 2x + \frac{1}{4}\)
d. \(\left(x^2 + \frac{2}{5}y\right)\left(x^2 - \frac{2}{5}y\right) = x^4 - \left(\frac{2}{5}y\right)^2 = x^4 - \frac{4}{25}y^2\)

2. Simplify:
a. \(\left(x + \frac{1}{3}\right)^3 = x^3 + x^2\left(\frac{1}{3}\right) + x\left(\frac{1}{3}\right)^2 + \left(\frac{1}{3}\right)^3 = x^3 + \frac{1}{3}x^2 + \frac{1}{9}x + \frac{1}{27}\)
b. \(\left(2x + \frac{1}{2}\right)^3 = 8x^3 + 3 \cdot 4x^2 \cdot \frac{1}{2} + 3 \cdot 2x \cdot \left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^3 = 8x^3 + 6x^2 + \frac{3}{2}x + \frac{1}{8}\)
c. \((x - 3y)(x^2 + 3xy + 9y^2) = x^3 + 3x^2y + 9xy^2 - 3yx^2 - 9xy^2 - 27y^3 = x^3 - 27y^3\)
d. \(\left(\frac{1}{3}x + 2y\right)\left(\frac{1}{9}x^2 - \frac{2}{3}xy + 4y^2\right) = \frac{1}{3}x \cdot \frac{1}{9}x^2 + \frac{1}{3}x \cdot \left(-\frac{2}{3}xy\right) + \frac{1}{3}x \cdot 4y^2 + 2y \cdot \frac{1}{9}x^2 + 2y \cdot \left(-\frac{2}{3}xy\right) + 2y \cdot 4y^2 = \frac{1}{27}x^3 - \frac{2}{9}x^2y + \frac{4}{3}xy^2 + \frac{2}{9}x^2y - \frac{4}{3}xy^2 + 8y^3 = \frac{1}{27}x^3 + 8y^3\)

These are the simplified forms of the given algebraic expressions.