Use algebra tiles to find $$ \left(-9 x^{2}-10 x-8\right)-\left(-5 x^{2}-4 x-6\right) $$

Question

UpStudy AI · Accepted Answer

To solve the expression $$ \left(-9x^{2} - 10x - 8\right) - \left(-5x^{2} - 4x - 6\right) $$ using **algebra tiles**, we'll break down each polynomial and perform the subtraction step by step.

### Understanding Algebra Tiles

Algebra tiles are a visual representation of algebraic expressions, where:

- **Squares** represent $$ x^{2} $$ terms.
- **Rectangles** represent $$ x $$ terms.
- **Small squares** represent constant terms.

Negative tiles are typically shown with different colors or marked with a minus sign.

### Step-by-Step Solution

1. **Represent Each Polynomial with Algebra Tiles:**

- **First Polynomial:** $$ -9x^{2} - 10x - 8 $$
     - **Squares:** 9 negative $$ x^{2} $$ tiles
     - **Rectangles:** 10 negative $$ x $$ tiles
     - **Small Squares:** 8 negative constant tiles

- **Second Polynomial:** $$ -5x^{2} - 4x - 6 $$
     - **Squares:** 5 negative $$ x^{2} $$ tiles
     - **Rectangles:** 4 negative $$ x $$ tiles
     - **Small Squares:** 6 negative constant tiles

2. **Set Up the Subtraction:**

The expression is:
   $$
   \left(-9x^{2} - 10x - 8\right) - \left(-5x^{2} - 4x - 6\right)
   $$
   Subtraction of the second polynomial is equivalent to adding its opposite:
   $$
   -9x^{2} - 10x - 8 + 5x^{2} + 4x + 6
   $$

3. **Combine Like Terms:**

- **$$ x^{2} $$ Terms:**
     $$
     -9x^{2} + 5x^{2} = (-9 + 5)x^{2} = -4x^{2}
     $$
   
   - **$$ x $$ Terms:**
     $$
     -10x + 4x = (-10 + 4)x = -6x
     $$
   
   - **Constant Terms:**
     $$
     -8 + 6 = -2
     $$

4. **Final Simplified Expression:**

Combining all the like terms gives:
   $$
   -4x^{2} - 6x - 2
   $$

### Visual Representation with Algebra Tiles

While I can't provide actual tiles here, imagine arranging the tiles as follows:

- Start with **9 negative $$ x^{2} $$ tiles**, **10 negative $$ x $$ tiles**, and **8 negative constant tiles**.
- **Subtracting** (or adding the opposite of) **5 negative $$ x^{2} $$ tiles**, **4 negative $$ x $$ tiles**, and **6 negative constant tiles** means you're effectively adding **5 positive $$ x^{2} $$ tiles**, **4 positive $$ x $$ tiles**, and **6 positive constant tiles**.
- After combining, you'll have **4 negative $$ x^{2} $$ tiles**, **6 negative $$ x $$ tiles**, and **2 negative constant tiles** left.

### Conclusion

Using algebra tiles helps visualize the subtraction by breaking down each term and combining like terms systematically. The simplified result of the given expression is:

$$
-4x^{2} - 6x - 2
$$
The simplified expression is $$ -4x^{2} - 6x - 2 $$.

Use algebra tiles to find

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Understanding Algebra Tiles

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