Find the values of p for which both the roots of the equation `4x^2 - 20px + (25p^2 + 15p - 66) = 0` are less than 2.

Answer

To find the values of \( p \) for which both roots of the equation

\[ 4x^2 - 20px + (25p^2 + 15p - 66) = 0 \]

are less than 2, we need to follow these steps:

### Step 1: Ensure the roots are real
The roots of the quadratic equation are real if the discriminant is non-negative. The discriminant \( D \) is given by:

\[ D = b^2 - 4ac \]

For our equation, \( a = 4 \), \( b = -20p \), and \( c = 25p^2 + 15p - 66 \). Thus, we have:

\[ D = (-20p)^2 - 4 \cdot 4 \cdot (25p^2 + 15p - 66) \]

Calculating this gives:

\[ D = 400p^2 - 16(25p^2 + 15p - 66) \]

Expanding the second term:

\[ D = 400p^2 - (400p^2 + 240p - 1056) \]

Simplifying this results in:

\[ D = 400p^2 - 400p^2 - 240p + 1056 = -240p + 1056 \]

Setting the discriminant \( D \geq 0 \):

\[ -240p + 1056 \geq 0 \]

### Step 2: Solve the inequality
Rearranging gives:

\[ 240p \leq 1056 \]

Dividing both sides by 240:

\[ p \leq \frac{1056}{240} = \frac{22}{5} \]

### Step 3: Ensure both roots are less than 2
Next, we need to ensure that both roots are less than 2. For this, we evaluate the quadratic at \( x = 2 \):

\[ f(2) = 4(2^2) - 20p(2) + (25p^2 + 15p - 66) \]

Calculating \( f(2) \):

\[ f(2) = 16 - 40p + (25p^2 + 15p - 66) \]

This simplifies to:

\[ f(2) = 25p^2 - 25p - 50 \]

Setting \( f(2) > 0 \):

\[ 25p^2 - 25p - 50 > 0 \]

Dividing through by 25:

\[ p^2 - p - 2 > 0 \]

### Step 4: Factor the quadratic
Factoring gives:

\[ (p - 2)(p + 1) > 0 \]

### Step 5: Determine intervals
The critical points are \( p = -1 \) and \( p = 2 \). We test the intervals:

1. \( p < -1 \): Choose \( p = -2 \) → \( (-2 - 2)(-2 + 1) = (-4)(-1) > 0 \)
2. \( -1 < p < 2 \): Choose \( p = 0 \) → \( (0 - 2)(0 + 1) = (-2)(1) < 0 \)
3. \( p > 2 \): Choose \( p = 3 \) → \( (3 - 2)(3 + 1) = (1)(4) > 0 \)

Thus, the solution to \( (p - 2)(p + 1) > 0 \) is:

\[ p < -1 \quad \text{or} \quad p > 2 \]

### Step 6: Combine results
From Step 2, we found \( p \leq \frac{22}{5} \). Therefore, we combine this with the intervals from Step 5:

1. \( p < -1 \)
2. \( p \leq \frac{22}{5} \) (which is approximately 4.4)

Since \( p > 2 \) does not satisfy \( p \leq \frac{22}{5} \), we only consider \( p < -1 \).

### Final Answer
Thus, the values of \( p \) for which both roots of the equation are less than 2 are:

\[ p < -1 \]