Let p(x)=6x^4+ax^3+bx^2+cx-20. It is given that x+4 is a factor of p(x) and p(x)=(dx^2+ex+4)(2x^2-x+f). Find the values of d, e and f.

To solve for the values of \( d \), \( e \), and \( f \), let's first utilize the given information systematically.

### Step 1: Use the factor theorem
Given that \( x + 4 \) is a factor of \( p(x) \), this means \( p(-4) = 0 \).

Substitute \( x = -4 \) into the polynomial:
\[
p(-4) = 6(-4)^4 + a(-4)^3 + b(-4)^2 + c(-4) - 20 = 0
\]
Calculate the powers of \(-4\):
\[
(-4)^4 = 256, \quad (-4)^3 = -64, \quad (-4)^2 = 16
\]
Substitute these into the equation:
\[
p(-4) = 6(256) + a(-64) + b(16) - 4c - 20 = 0
\]
\[
1536 - 64a + 16b - 4c - 20 = 0
\]
\[
64a - 16b + 4c = 1516 \quad \text{(Equation 1)}
\]

### Step 2: Expand the product of quadratic polynomials
Given \( p(x) = (dx^2 + ex + 4)(2x^2 - x + f) \), let's expand this product.

\[
p(x) = (dx^2 + ex + 4)(2x^2 - x + f)
\]
Multiply the terms:
\[
= d(2x^2 - x + f)x^2 + e(2x^2 - x + f)x + 4(2x^2 - x + f)
\]
Expand each multiplication:
\[
= 2dx^4 - dx^3 + dfx^2 + 2ex^3 - ex^2 + efx + 8x^2 - 4x + 4f
\]
Combine like terms:
\[
p(x) = 2dx^4 + (2d - 1)ex^3 + (df + 8 - e)x^2 + (ef - 4)x + 4f
\]
This must match the original polynomial \( p(x) = 6x^4 + ax^3 + bx^2 + cx - 20 \).

### Step 3: Equate coefficients
By equating the coefficients of \( x^4 \), \( x^3 \), \( x^2 \), \( x \), and the constant term:

- Coefficient of \( x^4 \): \( 2d = 6 \) → \( d = 3 \)
- Coefficient of \( x^3 \): \( 2e - d = a \) → \( 2e - 3 = a \)
- Coefficient of \( x^2 \): \( df + 8 - e = b \) → \( 3f + 8 - e = b \)
- Coefficient of \( x \): \( ef - 4 = c \) → \( 3f - 4 = c \)
- Constant term: \( 4f = -20 \) → \( f = -5 \)

### Step 4: Substitute \( f \) back into the equations
- From \( f = -5 \):
    - \( 3f = -15 \)
    - Therefore, \( c = -15 - 4 = -19 \)
    - \( b = 3(-5) + 8 - e = -15 + 8 - e = -7 - e \)
- \( a = 2e - 3 \)

### Step 5: Combine the results
- Given \( b = -7 - e \) and equation 1 is \( 64a - 16b + 4c = 1516 \), substitute \( a \), \( b \), and \( c \) into the equation:

Using the substitution:
- \( a = 2e - 3 \)
- \( b = -7 - e \)
- \( c = -19 \)

Substitute back:
\[
64(2e - 3) - 16(-7 - e) + 4(-19) = 1516
\]
Simplify and solve for \( e \):
\[
128e - 192 + 112 + 16e - 76 = 1516
\]
\[
144e - 156 = 1516
\]
\[
144e = 1672 \quad \Rightarrow \quad e = \frac{1672}{144} = 11.61
\]
Thus, the values of \( d \), \( e \), and \( f \) are:

\[
\boxed{d = 3}, \quad \boxed{e = 11.61}, \quad \boxed{f = -5}
\]

### Final Answer:
\[
\text{The values are } d = 3, e = 11.61, \text{ and } f = -5
\]
Do you want more details or have any questions?

1. How to use the factor theorem in other polynomial problems?
2. Can the factor theorem be used for polynomials of degrees higher than 4?
3. What happens if \( x - 4 \) was a factor instead of \( x + 4 \)?
4. How to factorize more complex polynomials?
5. What is the general approach to solving polynomial equations?

**Tip:** Always check your equations by substituting back into the original polynomial to verify the correctness of your solution.

Explore how to solve a polynomial equation using the factor theorem. Given that x + 4 is a factor of p(x) and p(x) can be expressed as the product of two quadratic polynomials, find the values of d, e, and f. Suitable for advanced high school students (Grades 10-12).

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