the equation 4(p-2x) = (12+15p)/(x+p) where p is constant has two distinct real roots. 
a) show that 3p^2 - 10p - 8 > 0 
b) using algebra find the range of possible values of p

Let's solve the given equation step by step.

### Part (a): Show that \( 3p^2 - 10p - 8 > 0 \)

We are given the equation:
\[
4(p - 2x) = \frac{12 + 15p}{x + p}
\]
where \( p \) is a constant, and the equation has two distinct real roots. First, we will manipulate the equation to derive the necessary inequality for \( p \).

#### Step 1: Multiply both sides by \( x + p \) to eliminate the denominator:
\[
4(p - 2x)(x + p) = 12 + 15p
\]

#### Step 2: Expand the left-hand side:
\[
4(p - 2x)(x + p) = 4(p(x + p) - 2x(x + p))
\]
\[
= 4(px + p^2 - 2x^2 - 2px)
\]
\[
= 4(p^2 - 2x^2 + px - 2px)
\]
\[
= 4(p^2 - 2x^2 - px)
\]

So the equation becomes:
\[
4(p^2 - 2x^2 - px) = 12 + 15p
\]

#### Step 3: Rearrange into standard quadratic form:
Now, we focus on the \( x \)-terms to write the equation in standard quadratic form \( ax^2 + bx + c = 0 \).

From the equation:
\[
4(-2x^2 - px + p^2) = 12 + 15p
\]
We can rewrite this as:
\[
-8x^2 - 4px + 4p^2 = 12 + 15p
\]
\[
-8x^2 - 4px + (4p^2 - 15p - 12) = 0
\]

Thus, the quadratic in \( x \) is:
\[
-8x^2 - 4px + (4p^2 - 15p - 12) = 0
\]
Now, the condition for two distinct real roots is that the discriminant of this quadratic must be positive. The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[
\Delta = b^2 - 4ac
\]

Here, \( a = -8 \), \( b = -4p \), and \( c = 4p^2 - 15p - 12 \).

#### Step 4: Calculate the discriminant:
\[
\Delta = (-4p)^2 - 4(-8)(4p^2 - 15p - 12)
\]
\[
= 16p^2 - 4(-8)(4p^2 - 15p - 12)
\]
\[
= 16p^2 + 32(4p^2 - 15p - 12)
\]
\[
= 16p^2 + 128p^2 - 480p - 384
\]
\[
= 144p^2 - 480p - 384
\]

For two distinct real roots, we require the discriminant \( \Delta > 0 \). Therefore, we need:
\[
144p^2 - 480p - 384 > 0
\]
Dividing the entire inequality by 48 to simplify:
\[
3p^2 - 10p - 8 > 0
\]
Thus, we have shown that \( 3p^2 - 10p - 8 > 0 \), which completes part (a).

---

### Part (b): Find the range of possible values of \( p \)

Now, we will solve the inequality \( 3p^2 - 10p - 8 > 0 \) to find the range of \( p \).

#### Step 1: Solve the quadratic equation \( 3p^2 - 10p - 8 = 0 \)

We use the quadratic formula:
\[
p = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(3)(-8)}}{2(3)}
\]
\[
p = \frac{10 \pm \sqrt{100 + 96}}{6}
\]
\[
p = \frac{10 \pm \sqrt{196}}{6}
\]
\[
p = \frac{10 \pm 14}{6}
\]
So the two roots are:
\[
p = \frac{10 + 14}{6} = 4 \quad \text{and} \quad p = \frac{10 - 14}{6} = -\frac{2}{3}
\]

#### Step 2: Analyze the inequality \( 3p^2 - 10p - 8 > 0 \)

The quadratic inequality \( 3p^2 - 10p - 8 > 0 \) is satisfied when \( p \) lies outside the roots \( p = -\frac{2}{3} \) and \( p = 4 \).

Thus, the solution to the inequality is:
\[
p < -\frac{2}{3} \quad \text{or} \quad p > 4
\]

### Final Answer:
The range of possible values of \( p \) is:
\[
p \in (-\infty, -\frac{2}{3}) \cup (4, \infty)
\]

Let me know if you'd like further details or explanations on any part of this solution!

### Related Questions:
1. How do you derive the quadratic discriminant formula?
2. What are the properties of quadratic inequalities?
3. What is the significance of having two distinct real roots for a quadratic equation?
4. Can the method used here be applied to non-quadratic equations?
5. How do different values of constants like \( p \) affect the nature of the roots?

**Tip:** When solving quadratic inequalities, always check the signs of intervals around the roots to determine where the inequality holds.

The equation 4(p - 2x) = (12 + 15p) / (x + p) where p is constant has two distinct real roots. 
 a) Show that 3p^2 - 10p - 8 > 0. 
 b) Using algebra, find the range of possible values of p.

This solution explains how to solve the equation 4(p - 2x) = (12 + 15p) / (x + p) where p is constant and the equation has two distinct real roots. It shows how to prove that 3p^2 - 10p - 8 > 0 and uses algebra to find the range of possible values for p. Explore key concepts such as discriminants, quadratic equations, and inequalities in this detailed step-by-step explanation.

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