Let's solve the equation step by step.

The given equation is:

$\frac{3}{2} \left( 4 + 12h \right) = \frac{1}{4} \left( 12 + 6h \right)$

Step 1: Distribute the fractions on both sides

For the left side:

$\frac{3}{2} \times 4 = 6, \quad \frac{3}{2} \times 12h = 18h$

So the left-hand side becomes:

$6 + 18h$

For the right side:

$\frac{1}{4} \times 12 = 3, \quad \frac{1}{4} \times 6h = \frac{6h}{4} = \frac{3h}{2}$

So the right-hand side becomes:

$3 + \frac{3h}{2}$

Now, the equation is:

$6 + 18h = 3 + \frac{3h}{2}$

Step 2: Eliminate the fraction by multiplying everything by 2

To get rid of the fraction $\frac{3h}{2}$, multiply the entire equation by 2:

$2 \times \left( 6 + 18h \right) = 2 \times \left( 3 + \frac{3h}{2} \right)$

This simplifies to:

$12 + 36h = 6 + 3h$

Step 3: Move all $h$-terms to one side

Subtract $3h$ from both sides:

$12 + 36h - 3h = 6$

Simplifies to:

$12 + 33h = 6$

Step 4: Isolate the $h$-term

Subtract 12 from both sides:

$33h = 6 - 12$

$33h = -6$

Step 5: Solve for $h$

Divide both sides by 33:

$h = \frac{-6}{33} = -\frac{2}{11}$

Thus, the solution is:

$h = -\frac{2}{11}$

Would you like a detailed explanation of any step?

Here are 5 related questions to expand your understanding:

1. How do you simplify an equation with fractions?
2. Why is it necessary to eliminate fractions by multiplying?
3. How do you check if