We can solve the equation step by step:

\[
-2x(32 - 3x) + 97 = 3(-1 + 5x)
\]

### Step 1: Distribute the terms on both sides.
On the left side, distribute \(-2x\) to both \(32\) and \(-3x\):

\[
-2x(32) = -64x \quad \text{and} \quad -2x(-3x) = 6x^2
\]

So, the left side becomes:

\[
6x^2 - 64x + 97
\]

On the right side, distribute \(3\) to \(-1\) and \(5x\):

\[
3(-1) = -3 \quad \text{and} \quad 3(5x) = 15x
\]

So, the right side becomes:

\[
-3 + 15x
\]

### Step 2: Write the equation after distribution:
\[
6x^2 - 64x + 97 = 15x - 3
\]

### Step 3: Move all terms to one side of the equation.
Subtract \(15x\) and add 3 to both sides:

\[
6x^2 - 64x - 15x + 97 + 3 = 0
\]

Simplify:

\[
6x^2 - 79x + 100 = 0
\]

### Step 4: Solve the quadratic equation.
We now have a quadratic equation:

\[
6x^2 - 79x + 100 = 0
\]

We can solve this using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where \(a = 6\), \(b = -79\), and \(c = 100\).

First, calculate the discriminant:

\[
\Delta = (-79)^2 - 4(6)(100) = 6241 - 2400 = 3841
\]

Now, plug the values into the quadratic formula:

\[
x = \frac{-(-79) \pm \sqrt{3841}}{2(6)}
\]

\[
x = \frac{79 \pm \sqrt{3841}}{12}
\]

Since \(\sqrt{3841} \approx 61.99\), we have:

\[
x = \frac{79 \pm 61.99}{12}
\]

### Step 5: Find the two solutions.
For the positive root:

\[
x = \frac{79 + 61.99}{12} = \frac{140.99}{12} \approx 11.75
\]

For the negative root:

\[
x = \frac{79 - 61.99}{12} = \frac{17.01}{12} \approx 1.42
\]

### Final solutions:
The solutions to the equation are approximately:

\[
x \approx 11.75 \quad \text{or} \quad x \approx 1.42
\]

Would you like further details or clarifications?

### Related Questions:
1. How do you verify the solution of a quadratic equation?
2. What is the discriminant, and how does it determine the nature of the roots?
3. How do you graph a quadratic equation like \(6x^2 - 79x + 100 = 0\)?
4. What are the properties of the quadratic formula?
5. How does changing the coefficients of a quadratic equation affect its graph?

**Tip:** Always check your solutions by substituting them back into the original equation.

Learn how to solve the quadratic equation -2x(32 - 3x) + 97 = 3(-1 + 5x) with a step-by-step solution. This problem involves algebraic concepts like distributing terms, simplifying, and using the quadratic formula. The final solutions are approximately x ≈ 11.75 or x ≈ 1.42. Suitable for students in Grades 9-11.

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