To find the product of the given expressions [tex]\(\left(\frac{3x + 5}{y^5}\right) \left(\frac{4y^3}{5}\right)\)[/tex], we will use the properties of fractions and multiplication. Here's a step-by-step solution:
1. Multiply the numerators:
The numerators are [tex]\(3x + 5\)[/tex] and [tex]\(4y^3\)[/tex].
[tex]\[
(3x + 5) \cdot 4y^3 = 4y^3(3x + 5) = 12xy^3 + 20y^3
\][/tex]
2. Multiply the denominators:
The denominators are [tex]\(y^5\)[/tex] and [tex]\(5\)[/tex].
[tex]\[
y^5 \cdot 5 = 5y^5
\][/tex]
3. Combine the results:
We can now write the product of the expressions as:
[tex]\[
\frac{12xy^3 + 20y^3}{5y^5}
\][/tex]
4. Simplify the fraction:
We will divide both terms in the numerator by the denominator:
[tex]\[
\frac{12xy^3}{5y^5} + \frac{20y^3}{5y^5}
\][/tex]
Simplify each term:
[tex]\[
\frac{12xy^3}{5y^5} = \frac{12x}{5y^2}
\][/tex]
[tex]\[
\frac{20y^3}{5y^5} = \frac{20}{5y^2} = \frac{4}{y^2}
\][/tex]
5. Combine the simplified terms:
[tex]\[
\frac{12x}{5y^2} + \frac{4}{y^2}
\][/tex]
6. Factor out the common term [tex]\(\frac{1}{y^2}\)[/tex]:
[tex]\[
\frac{12x + 4}{5y^2}
\][/tex]
Thus, the simplified form of the product of the expressions is:
[tex]\[
\frac{12x + 20}{5y^2}
\][/tex]
So the correct answer is:
[tex]\[
\boxed{\frac{12x + 20}{5y^2}}
\][/tex]
