Let's determine the numerator needed to make an equivalent fraction with the given denominator.
We start with the original fraction:
[tex]\[
\frac{4x}{5x - 2}
\][/tex]
We need to transform this into a fraction with the new denominator:
[tex]\[
\frac{\square}{x(5x - 2)(x + 5)}
\][/tex]
To achieve this, we need to find a new numerator that, when divided by the new denominator, remains equivalent to the original fraction.
First, let's write down what we know:
- The original numerator is \(4x\).
- The original denominator is \(5x - 2\).
- The new denominator is \(x(5x - 2)(x + 5)\).
Next, notice that the new denominator \(x(5x - 2)(x + 5)\) contains an extra factor of \(x\) and \((x + 5)\) compared to the original denominator \(5x - 2\). Therefore, to keep the fraction equivalent to the original, the numerator must also be multiplied by these extra factors.
Hence, we multiply the original numerator \(4x\) by the additional factors \(x\) and \(x + 5\):
[tex]\[
4x \cdot x \cdot (x + 5)
\][/tex]
Now perform the multiplication step-by-step:
[tex]\[
4x \cdot x = 4x^2
\][/tex]
[tex]\[
4x^2 \cdot (x + 5) = 4x^2(x + 5)
\][/tex]
Therefore, the numerator we need is:
[tex]\[
4x^2(x + 5)
\][/tex]
Thus, the filled in equivalent fraction is:
[tex]\[
\frac{4x}{5x - 2} = \frac{4x^2(x + 5)}{x(5x - 2)(x + 5)}
\][/tex]
