p*((-2b+√(4b^2-12a(d-p) ))/6a)-(a((-2b+√(4b^2-12a(d-p) ))/6a)^3  +b((-2b+√(4b^2-12a(d-p) ))/6a)^2+d((-2b+√(4b^2-12a(d-p) ))/6a)+c)

Let's break down and simplify the given expression step by step. The expression provided is:

\[
p \left(\frac{-2b + \sqrt{4b^2 - 12a(d - p)}}{6a}\right) - \left[a \left(\frac{-2b + \sqrt{4b^2 - 12a(d - p)}}{6a}\right)^3 + b \left(\frac{-2b + \sqrt{4b^2 - 12a(d - p)}}{6a}\right)^2 + d \left(\frac{-2b + \sqrt{4b^2 - 12a(d - p)}}{6a}\right) + c\right]
\]

### Step 1: Define the expression for simplicity
Let:
\[
x = \frac{-2b + \sqrt{4b^2 - 12a(d - p)}}{6a}
\]
So, the expression simplifies to:
\[
p \cdot x - \left[a x^3 + b x^2 + d x + c\right]
\]

### Step 2: Substitute \(x\) back into the equation
\[
p \cdot x - \left[a x^3 + b x^2 + d x + c\right]
\]

Substitute \( x \) back into the expression:

\[
= p \left(\frac{-2b + \sqrt{4b^2 - 12a(d - p)}}{6a}\right) - \left[a \left(\frac{-2b + \sqrt{4b^2 - 12a(d - p)}}{6a}\right)^3 + b \left(\frac{-2b + \sqrt{4b^2 - 12a(d - p)}}{6a}\right)^2 + d \left(\frac{-2b + \sqrt{4b^2 - 12a(d - p)}}{6a}\right) + c\right]
\]

### Step 3: Simplify the expression
The expression contains cubic, quadratic, linear terms in \( x \), and a constant \( c \). The simplified form is:

\[
= p \cdot x - \left[a x^3 + b x^2 + d x + c\right]
\]

Since this expression is already in its most simplified symbolic form, any further simplification would depend on specific values for the variables \( a \), \( b \), \( c \), \( d \), and \( p \). However, in general, it's important to note that the cubic polynomial term \( a x^3 + b x^2 + d x + c \) is fully expanded and represents a non-linear transformation of the variable \( x \).

Would you like more details on any particular part of this simplification? Here are some related questions that could help deepen your understanding:

1. How does substituting for \( x \) simplify the expression?
2. What is the significance of each term in the expanded form?
3. Can this equation be simplified further if specific values for \( a \), \( b \), \( c \), \( d \), and \( p \) are provided?
4. How does the expression relate to solving cubic equations?
5. Can this expression be solved numerically for specific values of the coefficients?
6. How would you graph this expression to analyze its behavior?
7. What are the real-world applications of such cubic equations?
8. How does the discriminant of the cubic polynomial affect the roots of the equation?

**Tip:** When dealing with complex algebraic expressions, breaking them down into simpler components or substituting variables can significantly ease the simplification process.

Learn how to simplify and analyze the cubic equation p*((-2b+√(4b^2-12a(d-p)))/6a)-(a((-2b+√(4b^2-12a(d-p)))/6a)^3 + b((-2b+√(4b^2-12a(d-p)))/6a)^2+d((-2b+√(4b^2-12a(d-p)))/6a)+c). This problem covers key algebraic concepts, including cubic equations, discriminant of cubic polynomials, and their applications. Suitable for students in Grades 10-12.

Simplify and Analyze the Cubic Equation p*((-2b+√(4b^2-12a(d-p)))/6a)-(a((-2b+√(4b^2-12a(d-p)))/6a)^3 + b((-2b+√(4b^2-12a(d-p)))/6a)^2+d((-2b+√(4b^2-12a(d-p)))/6a)+c)

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