We are asked to expand $(1 + x)^n (1 - 2x)^4$ and find the coefficient of $x$, given that the coefficient of $x^2$ is 54.

Step 1: Expand each factor using the binomial theorem

We use the binomial theorem to expand each factor separately.

Expansion of $(1 + x)^n$:

$(1 + x)^n = 1 + \binom{n}{1}x + \binom{n}{2}x^2 + \dots$ The relevant terms for us are up to $x^2$, so we write: $(1 + x)^n = 1 + nx + \frac{n(n-1)}{2}x^2 + \dots$

Expansion of $(1 - 2x)^4$:

$(1 - 2x)^4 = 1 + \binom{4}{1}(-2x) + \binom{4}{2}(-2x)^2 + \dots$ The first few terms are: $(1 - 2x)^4 = 1 - 8x + 24x^2 + \dots$

Step 2: Multiply the two expansions

Now, we multiply the expansions $(1 + x)^n$ and $(1 - 2x)^4$ to find the terms involving $x$ and $x^2$.

Constant term (coefficient of $x^0$):

$1 \times 1 = 1$

Coefficient of $x$:

There are two ways to get the $x$-term:

1. $1 \times (-8x) = -8x$
2. $nx \times 1 = nx$

Thus, the coefficient of $x$ is: $n - 8$

Coefficient of $x^2$:

There are three ways to get the $x^2$-term:

1. $1 \times 24x^2 = 24x^2$
2. $nx \times (-8x) = -8nx^2$
3. $\frac{n(n-1)}{2}x^2 \times 1 = \frac{n(n-1)}{2}x^2$

Thus, the coefficient of $x^2$ is: $24 - 8n + \frac{n(n-1)}{2}$ We are told that this coefficient is 54, so we set up the equation: $24 - 8n + \frac{n(n-1)}{2} = 54$ Multiply through by 2 to eliminate the fraction: $48 - 16n + n(n-1) = 108$ Simplifying the quadratic expression: $48 - 16n + n^2 - n = 108$ $n^2 - 17n + 48 = 108$ $n^2 - 17n - 60 = 0$ Now solve the quadratic equation using the quadratic formula: $n = \frac{-(-17) \pm \sqrt{(-17)^2 - 4(1)(-60)}}{2(1)}$ $n = \frac{17 \pm \sqrt{289 + 240}}{2}$ $n = \frac{17 \pm \sqrt{529}}{2}$ $n = \frac{17 \pm 23}{2}$ Thus, $n = \frac{17 + 23}{2} = 20$ or $n = \frac{17 - 23}{2} = -3$.

Since $n$ is a positive integer, we take $n = 20$.

Step 3: Find the coefficient of $x$

The coefficient of $x$ is $n - 8$. Substituting $n = 20$, we get: $\text{Coefficient of } x = 20 - 8 = 12$

Thus, the coefficient of $x$ is $\boxed{12}$.

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Would you like more details or clarifications? Here are some related questions:

1. How is the binomial theorem applied in expansions?
2. Why did we stop the expansion at $x^2$?
3. Can this method be applied to other polynomials?
4. How do we solve quadratic equations in general?
5. How does the quadratic formula work in solving the equation?

Tip: When solving quadratic equations, always check if you can factorize it before using the quadratic formula!