Answer :

Sure, let's simplify the given expression in a detailed step-by-step manner.

The given expression is:

[tex]\[ \left( \frac{x^2 + 6x - 7}{x^4 + 8x^3 + 7x^2} \right) \cdot 3x^2 \][/tex]

### Step 1: Factor the numerator and denominator

1. Factor the numerator [tex]\( x^2 + 6x - 7 \)[/tex]:

The numerator is a quadratic equation. To factor it, we look for two numbers that multiply to [tex]\(-7\)[/tex] and add to [tex]\(6\)[/tex]. These numbers are [tex]\(7\)[/tex] and [tex]\(-1\)[/tex].

So, we can factor the numerator as follows:

[tex]\[ x^2 + 6x - 7 = (x + 7)(x - 1) \][/tex]

2. Factor the denominator [tex]\( x^4 + 8x^3 + 7x^2 \)[/tex]:

The denominator has a common factor [tex]\(x^2\)[/tex], so we can factor it out first:

[tex]\[ x^4 + 8x^3 + 7x^2 = x^2 (x^2 + 8x + 7) \][/tex]

Next, we factor the quadratic expression [tex]\(x^2 + 8x + 7\)[/tex]. We look for two numbers that multiply to [tex]\(7\)[/tex] and add to [tex]\(8\)[/tex]. These numbers are [tex]\(7\)[/tex] and [tex]\(1\)[/tex].

So, we can factor it as follows:

[tex]\[ x^2 + 8x + 7 = (x + 7)(x + 1) \][/tex]

Combining everything, the denominator becomes:

[tex]\[ x^4 + 8x^3 + 7x^2 = x^2 (x + 7)(x + 1) \][/tex]

### Step 2: Write the expression with the factors

Now we can rewrite the original expression using these factors:

[tex]\[ \left( \frac{(x + 7)(x - 1)}{x^2 (x + 7)(x + 1)} \right) \cdot 3x^2 \][/tex]

### Step 3: Simplify the fraction

In the fraction, we can cancel out the common factor [tex]\((x + 7)\)[/tex] in the numerator and the denominator:

[tex]\[ \frac{(x - 1)}{x^2 (x + 1)} \][/tex]

### Step 4: Multiply by the remaining expression [tex]\(3x^2\)[/tex]

Now we need to multiply the remaining fraction by [tex]\(3x^2\)[/tex]:

[tex]\[ \left( \frac{(x - 1)}{x^2 (x + 1)} \right) \cdot 3x^2 \][/tex]

The [tex]\(x^2\)[/tex] in the numerator and denominator cancel each other out:

[tex]\[ \frac{(x - 1) \cdot 3x^2}{x^2 (x + 1)} = \frac{3(x - 1)}{x + 1} \][/tex]

### Final simplified expression

The simplified form of the original expression is:

[tex]\[ \boxed{\frac{3(x - 1)}{x + 1}} \][/tex]

Hence, the final answer after simplification is:

[tex]\[ \frac{3(x - 1)}{x + 1} \][/tex]