2x+5 x+4

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x^2+3x+2 x^2+3x+2

i keep getting

x+9
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(x+1)(x+2)

We can't tell by reading

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7x + 4

(2x+5)/(x^2+3x+2) - (x+4)/(x^2+3x+2)

= (2x+5-x-4)/(x^2+3x+2)
= (x+1) / (x+1)(x+2)
= 1/(x+2)

To simplify the expression, we need to combine the two fractions and find a common denominator.

The first step is to factorize the denominators of both fractions: x^2 + 3x + 2.

The factors of the quadratic x^2 + 3x + 2 are (x + 1) and (x + 2). Therefore, the common denominator for both fractions is (x + 1)(x + 2).

Now, let's rewrite both fractions with the common denominator:

(2x + 5)/(x^2 + 3x + 2) - (x + 4)/(x^2 + 3x + 2)

Now, since both fractions have the same denominator, we can combine them by subtracting the numerators while keeping the denominator the same:

(2x + 5 - (x + 4))/(x^2 + 3x + 2)

Simplifying the numerator:

(2x + 5 - x - 4)/(x^2 + 3x + 2)

Combining like terms:

(x + 1)/(x^2 + 3x + 2)

Factoring the quadratic in the denominator:

(x + 1)/[(x + 1)(x + 2)]

Simplifying the expression, we can cancel out the common factor of x + 1:

1/(x + 2)

Therefore, the simplified expression is:

1/(x + 2) or simply 1/(x + 2)