a. Consider the expression
[tex]\frac{2x+4}{x^2-6x}\cdot\frac{x^2-36}{4x+8}[/tex]Use the property
[tex]a^2-b^2=(a+b)(a-b)[/tex]and cancel out the common terms in the numerator and denominator to simplify.
[tex]\begin{gathered} \frac{2x+4}{x^2-6x}\cdot\frac{x^2-36}{4x+8}=\frac{2x+4}{x(x-6)}\cdot\frac{(x+6)(x-6)}{2(2x+4)} \\ =\frac{x+6}{2x} \end{gathered}[/tex]b. Consider the expression
[tex]\frac{x-3}{7x}\cdot\frac{3x^2}{x^2-2x-3}[/tex]Factorize and cancel out the common terms in the numerator and denominator to simplify.
[tex]\begin{gathered} \frac{x-3}{7x}\cdot\frac{3x^2}{x^2-2x-3}=\frac{x-3}{7}\cdot\frac{3x}{(x-3)(x+1)} \\ =\frac{3x}{7(x+1)} \end{gathered}[/tex]c. Consider the expression
[tex]\frac{x+2}{x}\div\frac{3x+6}{x^2}[/tex]Cross multiply and cancel out the common terms in the numerator and denominator to simplify.
[tex]\begin{gathered} \frac{x+2}{x}\div\frac{3x+6}{x^2}=\frac{x+2}{x}\cdot\frac{x^2}{3x+6} \\ =(x+2)\cdot\frac{x}{3(x+2)} \\ =\frac{x}{3} \end{gathered}[/tex]d. Consider the expression
[tex]\frac{2x}{x+2}\div\frac{x^2}{2x+4}[/tex]Cross multiply and cancel out the common terms in the numerator and denominator to simplify.
[tex]\begin{gathered} \frac{2x}{x+2}\div\frac{x^2}{2x+4}=\frac{2x}{x+2}\cdot\frac{2x+4}{x^2} \\ =\frac{2}{x+2}\cdot\frac{2(x+2)}{x} \\ =\frac{4}{x} \end{gathered}[/tex]