Use algebra tiles to find \( \left(10 x^{2}+6 x-6\right)+(x-5) \)

To add the polynomials \( (10x^{2} + 6x - 6) \) and \( (x - 5) \) using **algebra tiles**, follow these steps: ### 1. **Understand Algebra Tiles Basics** Algebra tiles are visual tools that represent different types of terms in a polynomial: - **Large squares** represent \( x^{2} \) terms. - **Rectangles** represent \( x \) terms. - **Small squares** represent constant terms. Positive terms are typically shown with solid tiles, while negative terms use tiles with a different shading or color. ### 2. **Represent Each Polynomial with Tiles** **First Polynomial: \( 10x^{2} + 6x - 6 \)** - **\( 10x^{2} \):** 10 large squares. - **\( 6x \):** 6 rectangles. - **\-6:** 6 small squares (negative). **Second Polynomial: \( x - 5 \)** - **\( x \):** 1 rectangle. - **\-5:** 5 small squares (negative). ### 3. **Combine Like Tiles** Now, place all the tiles together: - **\( x^{2} \) Tiles:** \( 10 + 0 = 10 \) large squares. - **\( x \) Tiles:** \( 6 + 1 = 7 \) rectangles. - **Constant Tiles:** \( -6 + (-5) = -11 \) small squares. ### 4. **Write the Combined Polynomial** After combining like terms, the resulting polynomial is: \[ 10x^{2} + 7x - 11 \] ### 5. **Visual Summary** Here's a visual breakdown: | Term Type | First Polynomial | Second Polynomial | Total | |-----------|-------------------|--------------------|-----------------| | \( x^{2} \) | 10 large squares | 0 | **10x²** | | \( x \) | 6 rectangles | 1 rectangle | **7x** | | Constants | -6 small squares | -5 small squares | **-11** | ### 6. **Final Answer** \[ (10x^{2} + 6x - 6) + (x - 5) = 10x^{2} + 7x - 11 \]