The problem asks us to use the binomial expansion to approximate the expression:

$\sqrt{\frac{1 + 4x}{1 - x}}$

and show that it is approximately:

$1 + \frac{5}{2}x - \frac{5}{8}x^2.$

Step-by-Step Solution:

1. Rewrite the expression: $\sqrt{\frac{1 + 4x}{1 - x}} = \left( \frac{1 + 4x}{1 - x} \right)^{1/2}$

2. Apply the binomial expansion: We'll expand the numerator and denominator separately, using binomial expansions for small $x$, and then combine them.

   Binomial expansion of $(1 + 4x)^{1/2}$:

   The binomial expansion for $(1 + y)^{1/2}$ is: $(1 + y)^{1/2} \approx 1 + \frac{1}{2}y - \frac{1}{8}y^2 \quad \text{for small } y.$ Here, $y = 4x$, so: $(1 + 4x)^{1/2} \approx 1 + \frac{1}{2}(4x) - \frac{1}{8}(4x)^2 = 1 + 2x - 2x^2.$

   Binomial expansion of $(1 - x)^{-1/2}$:

   Similarly, the binomial expansion for $(1 - y)^{-1/2}$ is: $(1 - y)^{-1/2} \approx 1 + \frac{1}{2}y + \frac{3}{8}y^2 \quad \text{for small } y.$ Here, $y = x$, so: $(1 - x)^{-1/2} \approx 1 + \frac{1}{2}x + \frac{3}{8}x^2.$

3. Combine the expansions: Now, multiply the two expansions we found: $\left( 1 + 2x - 2x^2 \right) \left( 1 + \frac{1}{2}x + \frac{3}{8}x^2 \right).$ Use the distributive property (multiplying each term):

   & \left( 1 + 2x - 2x^2 \right) \left( 1 + \frac{1}{2}x + \frac{3}{8}x^2 \right) = \\ & 1(1 + \frac{1}{2}x + \frac{3}{8}x^2) + 2x(1 + \frac{1}{2}x + \frac{3}{8}x^2) - 2x^2(1 + \frac{1}{2}x + \frac{3}{8}x^2). \end{aligned}$$ Expanding this out: $$\begin{aligned} & 1 + \frac{1}{2}x + \frac{3}{8}x^2 \\ & + 2x + x^2 + \frac{3}{4}x^3 \\ & - 2x^2 - x^3 - \frac{3}{4}x^4. \end{aligned}$$

4. Simplify the terms: Combine like terms (group powers of $x$):

   1 & + \left( \frac{1}{2}x + 2x \right) + \left( \frac{3}{8}x^2 + x^2 - 2x^2 \right) + O(x^3) \\ & = 1 + \frac{5}{2}x - \frac{5}{8}x^2. \end{aligned}$$

Thus, we have shown that:

$\sqrt{\frac{1 + 4x}{1 - x}} \approx 1 + \frac{5}{2}x - \frac{5}{8}x^2,$

which is the required result.

Would you like further clarification or details on any part of the solution?

Here are 5 related questions to explore further:

1. How does the binomial expansion work for negative or fractional powers?
2. How can we approximate higher-order terms in the expansion for greater accuracy?
3. In which contexts are binomial expansions commonly used in calculus and physics?
4. What is the radius of convergence for binomial series expansions?
5. How does the approximation change for large values of $x$?

Tip: When using binomial expansions, make sure the variable is within the range where the series converges, usually when the absolute value of $x$ is less than 1 for expansions like this.