Certainly! Let's multiply the given expressions step-by-step and simplify the result.
We start with the expression:
[tex]\[
\frac{45 a^3}{a^2 - 4} \cdot \frac{a^2 + 2a}{30 a^2}
\][/tex]
Step 1: Multiply the Numerators
The numerators of the two fractions are:
[tex]\[
45 a^3 \quad \text{and} \quad a^2 + 2a
\][/tex]
Multiplying these together:
[tex]\[
45 a^3 \cdot (a^2 + 2a) = 45 a^3(a^2 + 2a) = 45 a^5 + 90 a^4
\][/tex]
Step 2: Multiply the Denominators
The denominators of the fractions are:
[tex]\[
a^2 - 4 \quad \text{and} \quad 30 a^2
\][/tex]
Multiplying these together:
[tex]\[
(a^2 - 4) \cdot 30 a^2 = 30 a^2 (a^2 - 4) = 30 a^4 - 120 a^2
\][/tex]
Step 3: Combine the Results
Combining the multiplied numerators and denominators, we get:
[tex]\[
\frac{45 a^5 + 90 a^4}{30 a^4 - 120 a^2}
\][/tex]
Step 4: Simplify the Fraction
To simplify, we should look for common factors in the numerator and the denominator. Let's rewrite and factor both numerator and denominator:
Numerator: [tex]\(45 a^5 + 90 a^4 = 45 a^4 (a + 2)\)[/tex]
Denominator: [tex]\(30 a^4 - 120 a^2 = 30 a^2 (a^2 - 4) = 30 a^2 (a + 2)(a - 2)\)[/tex]
So the expression becomes:
[tex]\[
\frac{45 a^4 (a + 2)}{30 a^2 (a + 2)(a - 2)}
\][/tex]
We can cancel out common factors from the numerator and the denominator. Specifically, [tex]\(45 a^4\)[/tex] and [tex]\(30 a^2\)[/tex] share a common factor:
[tex]\[
\frac{3 \cdot 15 a^4 (a + 2)}{2 \cdot 15 a^2 (a + 2)(a - 2)} = \frac{3 a^2}{2 (a - 2)}
\][/tex]
Thus, the product in simplest form is:
[tex]\[
\frac{3 a^2}{2 (a - 2)}
\][/tex]
From the options given, this matches with:
[tex]\[
\frac{3 a^2}{2 a - 4}
\][/tex]
So, the simplest form of the product is:
[tex]\[
\boxed{\frac{3 a^2}{2 (a-2)}}
\][/tex]
