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The given equation is:

\[
\frac{x}{4x - 1} = \frac{6x + 5}{12x + 31}
\]

To solve for \(x\), follow these steps:

### Step 1: Cross-multiply
Cross-multiply the two fractions:

\[
x(12x + 31) = (6x + 5)(4x - 1)
\]

### Step 2: Expand both sides
Expand both sides of the equation:

- Left side: 
\[
x(12x + 31) = 12x^2 + 31x
\]

- Right side:
\[
(6x + 5)(4x - 1) = 6x \cdot 4x + 6x \cdot (-1) + 5 \cdot 4x + 5 \cdot (-1)
= 24x^2 - 6x + 20x - 5 = 24x^2 + 14x - 5
\]

So, the equation becomes:
\[
12x^2 + 31x = 24x^2 + 14x - 5
\]

### Step 3: Move all terms to one side
Bring all terms to one side to set the equation to 0:

\[
12x^2 + 31x - (24x^2 + 14x - 5) = 0
\]
Simplify the equation:
\[
12x^2 + 31x - 24x^2 - 14x + 5 = 0
\]
\[
-12x^2 + 17x + 5 = 0
\]

### Step 4: Solve the quadratic equation
Now, solve the quadratic equation:

\[
-12x^2 + 17x + 5 = 0
\]

You can either use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = -12\), \(b = 17\), and \(c = 5\).

### Step 5: Apply the quadratic formula

Substitute the values of \(a\), \(b\), and \(c\):

\[
x = \frac{-17 \pm \sqrt{17^2 - 4(-12)(5)}}{2(-12)}
\]
\[
x = \frac{-17 \pm \sqrt{289 + 240}}{-24}
\]
\[
x = \frac{-17 \pm \sqrt{529}}{-24}
\]
\[
x = \frac{-17 \pm 23}{-24}
\]

### Step 6: Solve for the two possible values of \(x\)

1. \(x = \frac{-17 + 23}{-24} = \frac{6}{-24} = -\frac{1}{4}\)
2. \(x = \frac{-17 - 23}{-24} = \frac{-40}{-24} = \frac{5}{3}\)

Thus, the solutions are:
\[
x = -\frac{1}{4} \quad \text{or} \quad x = \frac{5}{3}
\]

Would you like more details on any step or a specific explanation? 

Here are some questions for further exploration:
1. How do you handle quadratic equations in general form?
2. Why is cross-multiplication useful in solving rational equations?
3. What conditions should be considered when solving rational equations?
4. Can you derive the quadratic formula from its basic form?
5. How do you verify the solutions of a quadratic equation?

**Tip:** Always check for any extraneous solutions when solving rational equations by substituting the values of \(x\) back into the original equation.

Solve for x: x / (4x - 1) = (6x + 5) / (12x + 31)

Learn how to solve the rational equation x / (4x - 1) = (6x + 5) / (12x + 31) using cross-multiplication and the quadratic formula. This step-by-step solution is suitable for students in Grades 9-11, covering key algebraic concepts.

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