Solution:

Introduction: In this problem, we are required to find the values of x in the given equation.

Step 1: Expanding the equation
We can start by expanding the given equation to simplify it. The equation is:

7(x – 2p)2 + 5p2 = 35xp + 117p2

Expanding the left-hand side of the equation, we get:

7(x2 – 4px + 4p2) + 5p2 = 35xp + 117p2

Simplifying further, we get:

7x2 – 28px + 28p2 + 5p2 = 35xp + 117p2

7x2 – 35xp + 33p2 = 0

Step 2: Factoring the equation
Now, we can try to factor the equation to find the values of x. The equation is:

7x2 – 35xp + 33p2 = 0

We can factor out a common factor of 7, which gives us:

7(x2 – 5xp + 11/7 p2) = 0

Now, we can solve for x by setting each factor equal to zero:

x2 – 5xp + 11/7 p2 = 0

Step 3: Applying the quadratic formula
We can solve for x using the quadratic formula, which is:

x = (-b ± √(b2 – 4ac)) / 2a

In this case, a = 1, b = -5p, and c = 11/7 p2. Substituting these values into the formula, we get:

x = (5p ± √(25p2 – 4(1)(11/7 p2))) / 2

Simplifying, we get:

x = (5p ± √(25p2 – 44/7 p2)) / 2

x = (5p ± √(131/7 p2)) / 2

Thus, the values of x are:

x = (5p + √(131/7 p2)) / 2

x = (5p – √(131/7 p2)) / 2

Conclusion: Therefore, the values of x in the equation 7(x – 2p)2 + 5p2 = 35xp + 117p2 are (5p + √(131/7 p2)) / 2 and (5p – √(131/7 p2)) / 2.