Sure, let's analyze and simplify the given expression step-by-step.
We need to determine the product of:
[tex]\[ \frac{2a - 7}{a} \times \frac{3a^2}{2a^2 - 11a + 14} \][/tex]
### Step 1: Factorize and Simplify
1. Identify the factorization of the denominator [tex]\(2a^2 - 11a + 14\)[/tex].
To factorize [tex]\(2a^2 - 11a + 14\)[/tex], we look for two numbers whose product is [tex]\(2 \times 14 = 28\)[/tex] and whose sum is [tex]\(-11\)[/tex]. These numbers are [tex]\(-4\)[/tex] and [tex]\(-7\)[/tex].
Split the middle term:
[tex]\[ 2a^2 - 4a - 7a + 14 \][/tex]
Group the terms:
[tex]\[ (2a^2 - 4a) - (7a - 14) \][/tex]
Factor each group:
[tex]\[ 2a(a - 2) - 7(a - 2) \][/tex]
Factor out the common term [tex]\((a - 2)\)[/tex]:
[tex]\[ (2a - 7)(a - 2) \][/tex]
So, [tex]\( 2a^2 - 11a + 14 = (2a - 7)(a - 2) \)[/tex].
### Step 2: Rewrite the Expression
Rewrite the original product using the factorization:
[tex]\[ \frac{2a - 7}{a} \times \frac{3a^2}{(2a - 7)(a - 2)} \][/tex]
### Step 3: Simplify the Expression
Now, cancel common factors:
The [tex]\((2a - 7)\)[/tex] term in the numerator and denominator cancel out:
[tex]\[ \frac{2a - 7}{a} \times \frac{3a^2}{(2a - 7)(a - 2)} = \frac{1}{a} \times \frac{3a^2}{a - 2} = \frac{3a^2}{a(a - 2)} \][/tex]
Further simplifying:
[tex]\[ \frac{3a^2}{a(a - 2)} = \frac{3a}{a - 2} \][/tex]
### Step 4: Identify the Correct Option
The simplified product of:
[tex]\[ \frac{2a - 7}{a} \times \frac{3a^2}{(2a - 7)(a - 2)} \][/tex]
is:
[tex]\[ \frac{3a}{a - 2} \][/tex]
So, the answer is:
[tex]\[ \frac{3a}{a - 2} \][/tex]
