turn 15-33x-15x^2 - 2x^3 into (1-2x)[(x+4)^2-1]

To transform the expression \(15 - 33x - 15x^2 - 2x^3\) into the factored form \((1 - 2x)\left[(x + 4)^2 - 1\right]\), let's follow these steps:

### 1. Expand the Target Expression
First, let's expand the given target expression to see if it simplifies to the original expression.

\[
(1 - 2x)\left[(x + 4)^2 - 1\right]
\]

Start by expanding \((x + 4)^2\):

\[
(x + 4)^2 = x^2 + 8x + 16
\]

Subtract 1 from this result:

\[
(x + 4)^2 - 1 = x^2 + 8x + 15
\]

Now, distribute \((1 - 2x)\) over \((x^2 + 8x + 15)\):

\[
(1 - 2x)(x^2 + 8x + 15) = 1 \cdot (x^2 + 8x + 15) - 2x \cdot (x^2 + 8x + 15)
\]

\[
= x^2 + 8x + 15 - 2x(x^2 + 8x + 15)
\]

Expand the second term:

\[
- 2x(x^2 + 8x + 15) = -2x^3 - 16x^2 - 30x
\]

Now, combine all the terms:

\[
x^2 + 8x + 15 - 2x^3 - 16x^2 - 30x
\]

Simplify by combining like terms:

\[
-2x^3 - 15x^2 - 22x + 15
\]

Rearrange the terms:

\[
15 - 22x - 15x^2 - 2x^3
\]

This expression should match the original one, but notice the discrepancy in the linear term. Let's check the original expression you provided:

### 2. Compare with the Original Expression
Original expression: \(15 - 33x - 15x^2 - 2x^3\)

Target expression: \(15 - 22x - 15x^2 - 2x^3\)

It seems that the linear coefficient does not match. The discrepancy might suggest a different form or there was a mistake in either the provided expression or the target factorization.

### 3. Revising the Expressions
Given that expanding the expression \((1 - 2x)\left[(x + 4)^2 - 1\right]\) does not match the original polynomial exactly, the problem might need reevaluation or a different approach to matching coefficients.

### Conclusion
While the expanded form of \((1 - 2x)\left[(x + 4)^2 - 1\right]\) closely resembles the original polynomial, there is a difference in the coefficient of the linear term (\(-33x\) vs. \(-22x\)). This indicates that the original and target forms might differ slightly.

If you have any questions or want to explore alternative factorizations, feel free to ask!

Learn how to transform the polynomial expression 15 - 33x - 15x^2 - 2x^3 into the factored form (1-2x)[(x+4)^2-1]. This step-by-step solution covers polynomial factorization and expansion techniques. Suitable for high school students.

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