Answer :

Certainly! Let's work through multiplying and simplifying the given algebraic fractions step by step:

### Expression to Simplify

We have the expression:

[tex]\[
\frac{6c^2 + 13c - 63}{6c^2 - 17c + 7} \cdot \frac{2c^2 - 9c + 4}{12c + 54}
\][/tex]

### Step 1: Factor the Polynomials

1. Numerator of the first fraction: [tex]\(6c^2 + 13c - 63\)[/tex]

This factors into [tex]\((2c - 3)(3c + 21)\)[/tex].

2. Denominator of the first fraction: [tex]\(6c^2 - 17c + 7\)[/tex]

This factors into [tex]\((3c - 1)(2c - 7)\)[/tex].

3. Numerator of the second fraction: [tex]\(2c^2 - 9c + 4\)[/tex]

This factors into [tex]\((2c - 1)(c - 4)\)[/tex].

4. Denominator of the second fraction: [tex]\(12c + 54\)[/tex]

This is equivalent to [tex]\(6(2c + 9)\)[/tex].

### Step 2: Write the Fraction Multiplication

After factoring, the expression becomes:

[tex]\[
\frac{(2c - 3)(3c + 21)}{(3c - 1)(2c - 7)} \cdot \frac{(2c - 1)(c - 4)}{6(2c + 9)}
\][/tex]

### Step 3: Multiply the Fractions

Combine the fractions by multiplying their numerators together and their denominators together:

Numerator: [tex]\((2c - 3)(3c + 21)(2c - 1)(c - 4)\)[/tex]

Denominator: [tex]\((3c - 1)(2c - 7)6(2c + 9)\)[/tex]

### Step 4: Simplify

To simplify, we look for common factors in the numerator and denominator that can be canceled out. After doing that, you are left with:

[tex]\[
\frac{c - 4}{6}
\][/tex]

So, the simplified form of the product of these fractions is:

[tex]\[
\frac{c}{6} - \frac{2}{3}
\][/tex]

Therefore, the simplified expression is [tex]\(\frac{c}{6} - \frac{2}{3}\)[/tex].

And there we have the final answer!

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