(x^2+dx−2)(x+4)=x^3+11x^2+26x−8

In the equation given, d is a constant. If the equation is true for all values of x, what is the value of dd?

We are given the equation:

\[
(x^2 + dx - 2)(x + 4) = x^3 + 11x^2 + 26x - 8
\]

Our goal is to determine the value of \( d \), assuming the equation is true for all values of \( x \).

### Step 1: Expand the Left-Hand Side
First, let's expand the left-hand side expression \( (x^2 + dx - 2)(x + 4) \).

Using the distributive property, we have:

\[
(x^2 + dx - 2)(x + 4) = (x^2)(x + 4) + (dx)(x + 4) + (-2)(x + 4)
\]

Now, expand each term:

\[
x^2(x + 4) = x^3 + 4x^2
\]
\[
dx(x + 4) = dx^2 + 4dx
\]
\[
-2(x + 4) = -2x - 8
\]

Now combine all terms:

\[
x^3 + 4x^2 + dx^2 + 4dx - 2x - 8
\]

This simplifies to:

\[
x^3 + (4 + d)x^2 + (4d - 2)x - 8
\]

### Step 2: Compare with the Right-Hand Side
The right-hand side of the equation is:

\[
x^3 + 11x^2 + 26x - 8
\]

Now, let's compare the terms on both sides:

- The \( x^3 \) terms are already equal: \( x^3 \).
- For the \( x^2 \) terms, we have:

\[
(4 + d)x^2 = 11x^2
\]

This gives us the equation:

\[
4 + d = 11
\]

Solving for \( d \):

\[
d = 11 - 4 = 7
\]

- For the \( x \) terms, we have:

\[
(4d - 2)x = 26x
\]

Substitute \( d = 7 \) into this equation:

\[
(4(7) - 2)x = 26x
\]
\[
(28 - 2)x = 26x
\]
\[
26x = 26x
\]

This is true, so the value of \( d \) is consistent.

### Final Answer:
The value of \( d \) is \( 7 \).

Would you like further details or have any questions?

Here are 5 follow-up questions to consider:
1. What happens to the equation if \( d \) has a different value?
2. How can you solve similar problems involving unknown constants in algebraic equations?
3. How would this change if the equation involved polynomials of higher degree?
4. Can you factor the expanded form of the left-hand side back into its original components?
5. How do the roots of the equation change as the value of \( d \) changes?

**Tip**: When comparing two polynomials, match the coefficients of corresponding powers of \( x \) to set up equations for any unknown constants!

Explore how to solve a polynomial equation involving an unknown constant d. Learn the distributive property and coefficient comparison in polynomials. Suitable for high school students studying algebra and polynomial equations.

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