The expression you've provided is:

\[
(c-p)(5x^2 + 17x - 12) + (x - 2)
\]

Let's break it down:

1. \((c-p)(5x^2 + 17x - 12)\): This represents the multiplication of the binomial \((c-p)\) by the quadratic polynomial \(5x^2 + 17x - 12\).
2. \((x - 2)\): This is a linear binomial that will be added to the result.

### 1. Expand the first part:

To expand \((c-p)(5x^2 + 17x - 12)\), distribute \((c-p)\) to each term in the polynomial:

\[
(c-p) \times 5x^2 + (c-p) \times 17x + (c-p) \times (-12)
\]

This results in:

\[
5x^2(c-p) + 17x(c-p) - 12(c-p)
\]

This is equivalent to:

\[
5x^2c - 5x^2p + 17xc - 17xp - 12c + 12p
\]

So the expanded expression for \((c-p)(5x^2 + 17x - 12)\) is:

\[
5x^2c - 5x^2p + 17xc - 17xp - 12c + 12p
\]

### 2. Add the binomial \((x - 2)\):

Now add \((x - 2)\) to the expanded expression:

\[
5x^2c - 5x^2p + 17xc - 17xp - 12c + 12p + x - 2
\]

The full expression is:

\[
5x^2c - 5x^2p + 17xc - 17xp - 12c + 12p + x - 2
\]

This is the final simplified form of the expression.

Would you like to go further with this expression, such as evaluating it for specific values of \(c\) and \(p\), or factoring it?

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Here are 5 related questions you might consider:

1. How do you factor trinomials like \(5x^2 + 17x - 12\)?
2. What happens if you substitute specific values for \(c\) and \(p\) in the expression?
3. How can you solve quadratic equations like \(5x^2 + 17x - 12 = 0\)?
4. What are the roots of the quadratic polynomial \(5x^2 + 17x - 12\)?
5. How does the expression change if you switch the sign of the binomial \((c-p)\)?

**Tip:** When expanding expressions, make sure to distribute each term carefully to avoid mistakes in the final equation.

Learn how to expand and simplify the expression (c-p)(5x^2 + 17x - 12) + (x - 2) with detailed step-by-step instructions. This problem covers polynomial expansion and addition of binomials in algebraic expressions. Suitable for high school students.

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