Sure, let's solve the given expression step-by-step.
We are given the expression:
[tex]\[ \frac{x^2 + 8x}{x^2 + 8x + 7} \][/tex]
To simplify this expression, follow these steps:
1. Factorize the numerator and the denominator (if possible):
The numerator is [tex]\(x^2 + 8x\)[/tex]. We can factor out an [tex]\(x\)[/tex]:
[tex]\[ x^2 + 8x = x(x + 8) \][/tex]
The denominator is [tex]\(x^2 + 8x + 7\)[/tex]. We look for two numbers that multiply to 7 and add up to 8. Those numbers are 1 and 7. Thus, we can factorize the denominator as follows:
[tex]\[ x^2 + 8x + 7 = (x + 1)(x + 7) \][/tex]
So, the expression now looks like this:
[tex]\[ \frac{x(x + 8)}{(x+1)(x+7)} \][/tex]
2. Simplify the expression:
The numerator and denominator do not have any common factors, so we cannot cancel anything out. Therefore, the expression remains in its factorized form.
Thus, the simplified expression is:
[tex]\[ \frac{x(x + 8)}{(x+1)(x+7)} \][/tex]
And that’s the simplified form of the given expression.
