Let's solve the problem step by step. Here's the given expression:

$\frac{\left(-\frac{4}{5}\right)^2}{\left(-\frac{1}{4} - \frac{1}{2} \times \frac{3}{4}\right)}$

Step 1: Simplify the numerator

The numerator is $\left(-\frac{4}{5}\right)^2$. Squaring a negative fraction makes it positive:

$\left(-\frac{4}{5}\right)^2 = \frac{16}{25}.$

Step 2: Simplify the denominator

The denominator involves two terms: $-\frac{1}{4} - \frac{1}{2} \times \frac{3}{4}.$

Step 2.1: Simplify $\frac{1}{2} \times \frac{3}{4}$

Multiply the fractions: $\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}.$

Step 2.2: Subtract from $-\frac{1}{4}$

Now compute: $-\frac{1}{4} - \frac{3}{8}.$

To subtract, find a common denominator. The least common denominator of 4 and 8 is 8: $-\frac{1}{4} = -\frac{2}{8}.$

Thus: $-\frac{2}{8} - \frac{3}{8} = -\frac{5}{8}.$

The denominator is $-\frac{5}{8}$.

Step 3: Compute the division

Now divide the numerator by the denominator: $\frac{\frac{16}{25}}{-\frac{5}{8}}.$

Dividing by a fraction is the same as multiplying by its reciprocal: $\frac{16}{25} \div -\frac{5}{8} = \frac{16}{25} \times -\frac{8}{5}.$

Multiply the numerators and denominators: $\frac{16}{25} \times -\frac{8}{5} = \frac{16 \cdot -8}{25 \cdot 5} = \frac{-128}{125}.$

Final Answer:

$\boxed{-\frac{128}{125}}$

Would you like further clarifications or explanations?

Follow-Up Questions:

1. Can you simplify $\frac{-128}{125}$ into a mixed number?
2. How do you find the reciprocal of a fraction?
3. What is the significance of squaring a fraction like $\left(-\frac{4}{5}\right)^2$?
4. Can this problem be solved using a calculator, and if so, how do you input fractions?
5. What happens if one term in the denominator is zero?

Tip:

Always simplify each component of a complex fraction separately before combining them—it reduces errors!