Let’s solve this step by step:

---

(a) Find the hidden expression

We know the mean of the three expressions is $3x$, and the expressions given are:

1. $3x - 10$
2. $3x$
3. $?$ (hidden expression, call it $E$)

The mean is calculated as: $\text{Mean} = \frac{\text{Sum of expressions}}{\text{Number of expressions}}$

So: $3x = \frac{(3x - 10) + 3x + E}{3}$

1. Multiply through by 3 to eliminate the denominator: $3 \cdot 3x = (3x - 10) + 3x + E$ $9x = (3x - 10) + 3x + E$

2. Combine like terms: $9x = 6x - 10 + E$

3. Solve for $E$ (the hidden expression): $E = 9x - 6x + 10$ $E = 3x + 10$

The hidden expression is $3x + 10$.

---

(b) Write three expressions with a mean of $4x$.

To create three expressions with a mean of $4x$, the sum of the expressions must equal $3 \cdot 4x = 12x$.

We can choose any three expressions that add up to $12x$. For example:

1. $2x$
2. $4x$
3. $6x$

Check: $\frac{2x + 4x + 6x}{3} = \frac{12x}{3} = 4x$

Thus, the set of expressions could be:
$2x, 4x, 6x$.

---

(c) Find the mean value of $2x + 3$, $5x - 9$, and $5x + 12$.

1. First, sum the three expressions: $(2x + 3) + (5x - 9) + (5x + 12)$

Combine like terms: $2x + 5x + 5x + 3 - 9 + 12$ $12x + 6$

2. Divide by 3 to find the mean: $\text{Mean} = \frac{12x + 6}{3}$

Simplify: $\text{Mean} = 4x + 2$

The mean value is $4x + 2$.

---

Summary of Solutions:

(a) Hidden expression: $3x + 10$
(b) Set of expressions: $2x, 4x, 6x$
(c) Mean value: $4x + 2$

---

Let me know if you need further clarification! Here are 5 related questions to practice:

1. What is the mean of $4x - 7$, $6x + 3$, and $2x - 1$?
2. If the mean of $3x, 5x, ?$ is $4x$, what is the missing expression?
3. Find three expressions that have a mean of $5x + 4$.
4. Calculate the mean of $7x + 2, 3x - 8, 4x + 10$.
5. Solve for $x$ if the mean of $2x - 4, 3x + 2, x + 6$ equals $5x$.

Tip: When finding the mean, always add all expressions first and then divide by the number of expressions.