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]
