Given the expression:
[tex]\frac{5x^2+25x+20}{7x}[/tex]
Let's determine where each piece belongs to create a rational expression equivalent to the expression given.
To determine, where each piece belong, let's input each value and simplify.
First simplify the given expression
[tex]\begin{gathered} \frac{5x^2+25x+20}{7x} \\ \\ =\frac{5(x+1)(x+4)}{7x} \end{gathered}[/tex]
Thus, we have:
[tex]\frac{x^2+2x+1}{x-1}\cdot\frac{5x^2+15x-20}{7x^2+7}[/tex]
Let's simplify the expression above to verify if it is equivalent to the simplified expression of the given expression.
We have:
[tex]\begin{gathered} \frac{x^2+2x+1}{x-1}\cdot\frac{5x^2+15x-20}{7x^2+7} \\ \\ =\frac{(x+1)^2}{x-1}\cdot\frac{5(x-1)(x+4)}{7x^2+7} \\ \\ =\frac{(x+1)^2}{x-1}\cdot\frac{5(x-1)(x+4)}{7x(x^{}+1)} \\ \\ =\frac{(x+1)}{1}\cdot\frac{5(x+4)}{7x} \\ \\ =\frac{5(x+1)(x+4)}{7x} \end{gathered}[/tex]
The expressions are equivalent.
Therefore, the correct expression is:
[tex]\frac{x^2+2x+1}{x-1}\cdot\frac{5x^2+15x-20}{7x^2+7}[/tex]
The expression in the numerator = 5x² + 15x - 20
The expression in the denominator = x - 1
ANSWER:
[tex]\frac{x^2+2x+1}{x-1}\cdot\frac{5x^2+15x-20}{7x^2+7}[/tex]
