Multiply (4x)/(x-1)*(x^2-1)/(3x+3)

Solution:

Step 1: Simplify the numerator.

Step 1.1: Express 1 as a square.

Rewrite the expression using $1^2$: $\frac{4x}{x-1}\cdot \frac{x^2-1^2}{3x+3}$.

Step 1.2: Apply the difference of squares.

Utilize the difference of squares $a^2-b^2=(a+b)(a-b)$, where $a=x$ and $b=1$: $\frac{4x}{x-1}\cdot \frac{(x+1)(x-1)}{3x+3}$.

Step 2: Simplify terms.

Step 2.1: Extract the common factor from the denominator.

Step 2.1.1: Factor out 3 from $3x$.

$\frac{4x}{x-1}\cdot \frac{(x+1)(x-1)}{3(x)+3}$.

Step 2.1.2: Factor out 3 from 3.

$\frac{4x}{x-1}\cdot \frac{(x+1)(x-1)}{3(x)+3(1)}$.

Step 2.1.3: Factor out 3 completely.

$\frac{4x}{x-1}\cdot \frac{(x+1)(x-1)}{3(x+1)}$.

Step 2.2: Eliminate the common $x-1$ factor.

Step 2.2.1: Factor $x-1$ from $(x+1)(x-1)$.

$\frac{4x}{x-1}\cdot \frac{(x-1)(x+1)}{3(x+1)}$.

Step 2.2.2: Cancel out the common factor.

$\frac{4x}{\cancel{x-1}}\cdot \frac{(\cancel{x-1})(x+1)}{3(x+1)}$.

Step 2.2.3: Rewrite the simplified expression.

$4x\cdot \frac{x+1}{3(x+1)}$.

Step 2.3: Merge 4 with $\frac{x+1}{3(x+1)}$.

$x\cdot \frac{4(x+1)}{3(x+1)}$.

Step 2.4: Combine $x$ with $\frac{4(x+1)}{3(x+1)}$.

$\frac{x(4(x+1))}{3(x+1)}$.

Step 2.5: Cancel the common $x+1$ factor.

Step 2.5.1: Cancel the common factor.

$\frac{x(4(\cancel{x+1}))}{3(\cancel{x+1})}$.

Step 2.5.2: Rewrite the simplified expression.

$\frac{x\cdot 4}{3}$.

Step 2.6: Rearrange $4$ to precede $x$.

$\frac{4x}{3}$.

Knowledge Notes:

1. Difference of Squares: This is a formula used to factor expressions of the form $a^2-b^2$ into $(a+b)(a-b)$. It is applicable when both terms are perfect squares.

2. Factoring: The process of breaking down an expression into its constituent factors. This can simplify expressions and is particularly useful for canceling out common factors in fractions.

3. Simplifying Fractions: When simplifying fractions, any common factors in the numerator and denominator can be canceled out. This process often involves factoring both the numerator and the denominator to identify common factors.

4. Common Factors: These are factors that are shared by two or more terms or expressions. Identifying and canceling common factors is a key step in simplifying expressions.

5. Rearranging Terms: Sometimes, terms within an expression can be rearranged for clarity or to match a conventional format, such as placing numerical coefficients before variables.