Questions: Multiply and simplify. [ (27 a-18 a^2)/(4 a^2+12 a+9) cdot (4 a^2+12 a+9)/(4 a^2-9) ]

Solution Steps

To solve the given problem, we need to follow these steps:

1. Factorize the numerators and denominators of both fractions.
2. Cancel out the common factors in the numerator and the denominator.
3. Simplify the resulting expression.

We start with the expression:

\[ \frac{27a - 18a^2}{4a^2 + 12a + 9} \cdot \frac{4a^2 + 12a + 9}{4a^2 - 9} \]

First, we factor the numerator and denominator of each fraction:

• The numerator \(27a - 18a^2\) can be factored as: \[ -9a(2a - 3) \]

• The denominator \(4a^2 + 12a + 9\) can be factored as: \[ (2a + 3)^2 \]

• The second numerator \(4a^2 + 12a + 9\) is the same as the first denominator, which is: \[ (2a + 3)^2 \]

• The denominator \(4a^2 - 9\) can be factored as: \[ (2a - 3)(2a + 3) \]

Now substituting the factored forms back into the expression, we have:

\[ \frac{-9a(2a - 3)}{(2a + 3)^2} \cdot \frac{(2a + 3)^2}{(2a - 3)(2a + 3)} \]

Next, we can cancel the common factors \((2a + 3)^2\) in the numerator and denominator:

\[ \frac{-9a(2a - 3)}{(2a - 3)(2a + 3)} \]

Now, we can cancel \((2a - 3)\) from the numerator and denominator (assuming \(2a - 3 \neq 0\)):

\[ \frac{-9a}{2a + 3} \]

Final Answer