To solve the inequality \( x(-5 - 3x) \leq 2x - 1 \), let's go step by step:

### Step 1: Expand the left-hand side
We distribute \(x\) over \(-5 - 3x\):

\[
x(-5 - 3x) = -5x - 3x^2
\]

So, the inequality becomes:

\[
-5x - 3x^2 \leq 2x - 1
\]

### Step 2: Move all terms to one side
We want to move all terms to one side of the inequality to simplify it. Subtract \(2x\) and add \(1\) from both sides:

\[
-5x - 3x^2 - 2x + 1 \leq 0
\]

Simplifying the left-hand side:

\[
-3x^2 - 7x + 1 \leq 0
\]

### Step 3: Solve the quadratic inequality
We now need to solve:

\[
-3x^2 - 7x + 1 \leq 0
\]

First, solve the corresponding quadratic equation:

\[
-3x^2 - 7x + 1 = 0
\]

We can use the quadratic formula:

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

where \(a = -3\), \(b = -7\), and \(c = 1\). Plugging in the values:

\[
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(-3)(1)}}{2(-3)}
\]
\[
x = \frac{7 \pm \sqrt{49 + 12}}{-6}
\]
\[
x = \frac{7 \pm \sqrt{61}}{-6}
\]

So, the solutions are:

\[
x = \frac{7 + \sqrt{61}}{-6} \quad \text{or} \quad x = \frac{7 - \sqrt{61}}{-6}
\]

We can approximate these values:

\[
x_1 \approx -2.434, \quad x_2 \approx 0.067
\]

### Step 4: Analyze the quadratic inequality
Now we need to determine where the quadratic expression is less than or equal to zero. Since the parabola opens downwards (because the coefficient of \(x^2\) is negative), the inequality \( -3x^2 - 7x + 1 \leq 0 \) holds between the two roots. Therefore, the solution to the inequality is:

\[
\boxed{x \in \left[ \frac{7 - \sqrt{61}}{-6}, \frac{7 + \sqrt{61}}{-6} \right]}
\]
or approximately:
\[
x \in [-2.434, 0.067]
\]

### Conclusion
The inequality \( x(-5 - 3x) \leq 2x - 1 \) has solutions in the interval \( x \in [-2.434, 0.067] \).

Would you like further details or clarifications?

Here are five related questions to deepen your understanding:
1. How do you solve quadratic inequalities in general?
2. What does the quadratic formula represent geometrically?
3. How can we graphically interpret the solutions of quadratic inequalities?
4. How do we verify solutions for quadratic inequalities?
5. How do the signs of the quadratic coefficients affect the parabola's shape?

**Tip:** When solving inequalities, always remember to check if you need to flip the inequality sign, especially when multiplying or dividing by negative numbers.

Explore the step-by-step process to solve the quadratic inequality x(-5 - 3x) ≤ 2x - 1. This detailed guide covers expanding expressions, using the quadratic formula, and analyzing quadratic inequalities, making it ideal for students in Grades 9-11.

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