Use algebra tiles to find $$ \left(-10 x^{2}+4 x+7\right)-\left(2 x^{2}+3 x\right) $$

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To solve the expression $$ \left(-10x^{2} + 4x + 7\right) - \left(2x^{2} + 3x\right) $$ using **algebra tiles**, we'll break down the process step by step. Algebra tiles are a visual tool that help in understanding and performing algebraic operations by representing different types of terms with specific tiles.

### Step 1: Represent Each Polynomial with Algebra Tiles

1. **First Polynomial: $$ -10x^{2} + 4x + 7 $$**
   - **$$ -10x^{2} $$**: Use **10 negative squared term tiles** (each representing $$ -x^{2} $$).
   - **$$ +4x $$**: Use **4 positive linear tiles** (each representing $$ +x $$).
   - **$$ +7 $$**: Use **7 positive constant tiles** (each representing $$ +1 $$).

2. **Second Polynomial: $$ 2x^{2} + 3x $$**
   - **$$ 2x^{2} $$**: Use **2 positive squared term tiles** (each representing $$ +x^{2} $$).
   - **$$ +3x $$**: Use **3 positive linear tiles** (each representing $$ +x $$).

### Step 2: Subtract the Second Polynomial

Subtracting the second polynomial is equivalent to **adding the additive inverses** of its terms:
- $$ -(2x^{2}) = -2x^{2} $$
- $$ -(3x) = -3x $$

So, we'll add:
- **$$ -2x^{2} $$**: Use **2 negative squared term tiles**.
- **$$ -3x $$**: Use **3 negative linear tiles**.

### Step 3: Combine All Tiles

Now, combine all tiles from the first polynomial and the subtracted second polynomial:

- **Squared Terms**:
  - Existing: **-10x²**
  - Subtracted: **-2x²**
  - **Total Squared Terms**: $$ -10x^{2} - 2x^{2} = -12x^{2} $$

- **Linear Terms**:
  - Existing: **+4x**
  - Subtracted: **-3x**
  - **Total Linear Terms**: $$ 4x - 3x = +x $$

- **Constant Terms**:
  - Existing: **+7**
  - No constants in the second polynomial to subtract.
  - **Total Constants**: **+7**

### Step 4: Write the Final Expression

After combining like terms, the resulting polynomial is:

$$
-12x^{2} + x + 7
$$

### Visual Summary with Algebra Tiles

While it's challenging to display actual tiles in text, here's a simplified representation:

```
First Polynomial:
-10x² | +4x | +7

Subtract Second Polynomial:
-2x² | -3x

Combined:
(-10x² - 2x²) + (4x - 3x) + 7 = -12x² + x + 7
```

### Final Answer

$$
-12x^{2} + x + 7
$$
$$ -12x^{2} + x + 7 $$

Use algebra tiles to find \( \left(-10 x^{2}+4 x+7\right)-\left(2 x^{2}+3 x\right) \)

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