1. Let's start by understanding some basic algebraic concepts. 2. **Algebraic manipulation** involves rearranging and simplifying expressions using operations like addition, subtraction, multiplication, division, and factoring. 3. A **term** is a single number, variable, or numbers and variables multiplied together, e.g., $3x$, $-5$, or $7xy$. 4. **Factors** are quantities multiplied together to get a product. For example, in $3x$, 3 and $x$ are factors. 5. A **polynomial** is an expression made up of terms added or subtracted, like $2x^2 + 3x - 5$. 6. Types of polynomials by number of terms: - Monomial: 1 term (e.g., $4x$) - Binomial: 2 terms (e.g., $x + 5$) - Trinomial: 3 terms (e.g., $x^2 + 3x + 2$) 7. **Algebra tiles** are visual tools representing terms: unit squares for constants, rectangles for variables, and larger rectangles for squared terms. 8. For example, a **unit square** represents 1, a **rectangular tile** might represent $x$, and a larger square tile represents $x^2$. 9. Now, let's solve a polynomial expression by factoring: Factor $x^2 + 5x + 6$. 10. Step 1: Find two numbers that multiply to 6 (constant term) and add to 5 (coefficient of $x$). 11. These numbers are 2 and 3 because $2 \times 3 = 6$ and $2 + 3 = 5$. 12. Step 2: Write the factored form as $(x + 2)(x + 3)$. 13. Step 3: Verify by expanding: $$ (x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6 $$ 14. This confirms the factorization is correct. 15. Algebraic manipulation and understanding terms, factors, and polynomials help simplify and solve equations effectively.