To simplify this mathematical expression, we first need to understand the structure of the expressions that we're given.
We have two fractions:
Expression 1: (-2x - 10) / (x^2 + 7x + 10)
Expression 2: 10 / (x + 2)
We're being asked to add these two fractions and simplify the result. Now, in order to add fractions, we must have a common denominator.
Looking at the denominators, you can see that the denominator in Expression 1 (x^2 + 7x + 10) is a quadratic expression that can be factored into two binomials.
Let's do that:
x^2 + 7x + 10 = (x + 2)(x + 5)
Now Expression 1 becomes: (-2x - 10) / [(x + 2)(x + 5)]
Comparing the denominators of the two expressions, we observe that the denominator (x+2) is common to both fractions.
Now, let's rewrite our problem:
(-2x - 10) / [(x + 2)(x + 5)] + 10 / (x + 2)
With the common denominator of (x + 2), we can add these two fractions:
{(-2x - 10) + 10*(x + 5)} / [(x + 2)(x + 5)]
Simplify the numerator:
(-2x - 10 + 10x + 50) / [(x + 2)(x + 5)]
Further simplification gives:
(8x + 40) / [(x + 2)(x + 5)]
The last step to simplifying fractions is to cancel all common factors from the numerator and the denominator.
We see that the number 8 is a common factor, let's divide both numerator and denominator by 8:
x + 5 / [(x + 2)(x + 5/8)]
After simplifying, the (x + 5) factor cancels out:
8 / (x + 2)
Hence, the simplified form of the given expression is 8 / (x + 2).