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

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UpStudy AI · Accepted Answer

Sure! Let's use **algebra tiles** to visually understand how to simplify the expression:

$$
\left(-6x^{2} + 10x - 10\right) + \left(9x^{2} + 10\right)
$$

### Step 1: Understand the Components

First, let's break down each part of the expression:

1. **First Polynomial:**
   - $$-6x^{2}$$: Six **negative** $$x^{2}$$ tiles.
   - $$+10x$$: Ten **positive** $$x$$ tiles.
   - $$-10$$: Ten **negative** unit tiles.

2. **Second Polynomial:**
   - $$+9x^{2}$$: Nine **positive** $$x^{2}$$ tiles.
   - $$+10$$: Ten **positive** unit tiles.

### Step 2: Combine Like Terms Using Algebra Tiles

**Algebra tiles** help us visualize the addition by grouping similar tiles together.

1. **Combine $$x^{2}$$ Tiles:**
   - **Negative $$x^{2}$$ Tiles:** $$-6x^{2}$$ means six negative $$x^{2}$$ tiles.
   - **Positive $$x^{2}$$ Tiles:** $$+9x^{2}$$ means nine positive $$x^{2}$$ tiles.
   
   **Combining:**
   $$
   (-6) + 9 = 3
   $$
   
   So, $$ -6x^{2} + 9x^{2} = 3x^{2} $$ (Three positive $$x^{2}$$ tiles remain).

2. **Combine $$x$$ Tiles:**
   - Only one set: $$+10x$$ (Ten positive $$x$$ tiles).

3. **Combine Unit Tiles:**
   - **Negative Unit Tiles:** $$-10$$ means ten negative unit tiles.
   - **Positive Unit Tiles:** $$+10$$ means ten positive unit tiles.
   
   **Combining:**
   $$
   (-10) + 10 = 0
   $$
   
   So, the unit tiles cancel each other out, leaving **no unit tiles**.

### Step 3: Write the Simplified Expression

After combining the like terms, the simplified expression is:

$$
3x^{2} + 10x
$$

### Visual Summary with Algebra Tiles

- **$$x^{2}$$ Tiles:** Start with $$-6x^{2}$$ and add $$+9x^{2}$$ → **$$3x^{2}$$**
- **$$x$$ Tiles:** Only $$+10x$$ remains.
- **Unit Tiles:** $$-10$$ and $$+10$$ cancel out → **0 units**

Thus, the final simplified expression is:

$$
\boxed{3x^{2} + 10x}
$$
The simplified expression is $$3x^{2} + 10x$$.

)) Use algebra tiles to find \( \left(-6 x^{2}+10 x-10\right)+\left(9 x^{2}+10\right) \)

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