Sure, let's perform and simplify the given operation step by step. We have to divide the fraction [tex]\(\frac{x^2 + 4x - 5}{2x^2}\)[/tex] by the fraction [tex]\(\frac{x-1}{4x}\)[/tex].
### Step 1: Rewrite Division as Multiplication by the Reciprocal
Recall that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite the problem as:
[tex]\[
\frac{x^2 + 4x - 5}{2x^2} \div \frac{x-1}{4x} = \frac{x^2 + 4x - 5}{2x^2} \times \frac{4x}{x-1}
\][/tex]
### Step 2: Multiply the Numerators Together
Multiply the numerators of both fractions:
[tex]\[
(x^2 + 4x - 5) \times (4x)
\][/tex]
[tex]\[
= 4x(x^2 + 4x - 5)
\][/tex]
Thus, the numerator of the resulting fraction is:
[tex]\[
4x(x^2 + 4x - 5)
\][/tex]
### Step 3: Multiply the Denominators Together
Multiply the denominators of both fractions:
[tex]\[
(2x^2) \times (x-1)
\][/tex]
[tex]\[
= 2x^2(x-1)
\][/tex]
Thus, the denominator of the resulting fraction is:
[tex]\[
2x^2(x-1)
\][/tex]
So now, our new fraction after performing the division is:
[tex]\[
\frac{4x(x^2 + 4x - 5)}{2x^2(x-1)}
\][/tex]
### Step 4: Simplify the Resulting Fraction
To simplify the fraction, divide both the numerator and the denominator by the common terms if any. Here, we can simplify:
[tex]\[
\frac{4x (x^2 + 4x - 5)}{2x^2(x - 1)}
\][/tex]
[tex]\[
= \frac{4x}{2x^2} \times \frac{(x^2 + 4x - 5)}{x-1}
\][/tex]
Further simplifying:
[tex]\[
= \frac{2}{x} \times \frac{(x^2 + 4x - 5)}{x - 1}
\][/tex]
We see that this cannot be simplified further by common factor cancellation, so we simplify the expression by dividing numerator and denominator where possible.
### Result:
Thus, the simplified result of the given operation is:
[tex]\[
\frac{4x(x^2 + 4x - 5)}{2x^2(x - 1)} = 2 + \frac{10}{x}
\][/tex]
So, the fraction simplifies to:
[tex]\(\boxed{2 + \frac{10}{x}}\)[/tex]
