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The problem provided asks to find the value of the expression \( px^2 + (m+n)x + n \), given the equation \( px^2 + (m+n)x + n = (5x + 3)(-2x + 7) \).

### Step 1: Expand the right-hand side of the equation

Expand \( (5x + 3)(-2x + 7) \) using the distributive property (FOIL method):

\[
(5x + 3)(-2x + 7) = 5x(-2x) + 5x(7) + 3(-2x) + 3(7)
\]

This simplifies as follows:

\[
= -10x^2 + 35x - 6x + 21
\]

Combine like terms:

\[
= -10x^2 + 29x + 21
\]

### Step 2: Compare coefficients

Now compare the expanded form \( -10x^2 + 29x + 21 \) to the given expression \( px^2 + (m+n)x + n \):

1. \( p = -10 \) (coefficient of \( x^2 \))
2. \( m+n = 29 \) (coefficient of \( x \))
3. \( n = 21 \) (constant term)

### Step 3: Solve for \( m \)

Since \( m + n = 29 \) and \( n = 21 \), substitute \( n = 21 \) into \( m + n = 29 \):

\[
m + 21 = 29
\]

\[
m = 8
\]

### Final Answer

The values of the constants are:
- \( p = -10 \)
- \( m = 8 \)
- \( n = 21 \)

Would you like further details or clarification on any step?

Here are some follow-up questions to consider:
1. How do we expand binomials using the distributive property?
2. Can you explain how coefficients of like terms are compared?
3. What if the expression was given in terms of different variables?
4. How would this approach change if we were solving a cubic equation?
5. Why is it important to combine like terms when expanding expressions?

**Tip:** Always compare corresponding coefficients when matching terms from both sides of a polynomial equation.

Find the value of the expression px^2 + (m+n)x + n, given that px^2 + (m+n)x + n = (5x + 3)(-2x + 7).

This problem involves finding the values of constants in a quadratic expression given in terms of binomial products. Learn how to expand and simplify expressions, compare coefficients, and solve for unknowns. Suitable for students in Grades 9-11.

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