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Simplify š‘„+4

š‘„āˆ’4
āˆ’
5š‘„+20
š‘„
2 āˆ’16Simplify š‘„+4
š‘„āˆ’4
āˆ’
5š‘„+20
š‘„
2 āˆ’16

To simplify the expression š‘„+4 / š‘„āˆ’4 - (5š‘„+20 / š‘„^2 -16), we need to find a common denominator.

The first fraction already has a common denominator of (š‘„āˆ’4), so we don't need to do anything with that term.

For the second fraction, we can factor the denominator as (š‘„+4)(š‘„-4). Therefore, the common denominator for both fractions is (š‘„+4)(š‘„āˆ’4).

Now, let's rewrite the fractions with the common denominator:
š‘„+4 / š‘„āˆ’4 = (š‘„+4)(š‘„+4) / (š‘„āˆ’4)(š‘„+4)
5š‘„+20 / š‘„^2 -16 = 5(š‘„+4) / (š‘„āˆ’4)(š‘„+4)

Now, we can combine the fractions:
(š‘„+4)(š‘„+4) / (š‘„āˆ’4)(š‘„+4) - 5(š‘„+4) / (š‘„āˆ’4)(š‘„+4)

To subtract the fractions, we need to have a common denominator, which we already have. So, we can combine the numerators:
(š‘„+4)(š‘„+4) - 5(š‘„+4) / (š‘„āˆ’4)(š‘„+4)

Next, we can simplify the numerator:
(š‘„+4)(š‘„+4) - 5(š‘„+4) = š‘„^2 + 8š‘„ + 16 - 5š‘„ - 20

Now, let's combine like terms in the numerator:
š‘„^2 + 8š‘„ + 16 - 5š‘„ - 20 = š‘„^2 + 3š‘„ - 4

Now, our simplified expression is:
(š‘„^2 + 3š‘„ - 4) / (š‘„āˆ’4)(š‘„+4)

To simplify the expression š‘„+4/š‘„āˆ’4āˆ’5š‘„+20/š‘„^2āˆ’16, we can follow these steps:

Step 1: Factor the denominators š‘„^2āˆ’16 and š‘„āˆ’4
The denominator š‘„^2āˆ’16 is a difference of squares and can be factored as (š‘„+4)(š‘„āˆ’4).
The denominator š‘„āˆ’4 is already factored.

Step 2: Find the least common denominator (LCD) of the fractions.
The LCD of the fractions is (š‘„+4)(š‘„āˆ’4).

Step 3: Rewrite each fraction with the LCD.
The first fraction š‘„+4/š‘„āˆ’4 can be rewritten as (š‘„+4)(š‘„+4)/(š‘„+4)(š‘„āˆ’4).
The second fraction āˆ’5š‘„+20/š‘„^2āˆ’16 can be rewritten as -5(š‘„āˆ’4)/(š‘„+4)(š‘„āˆ’4).

Step 4: Simplify the numerators.
The first fraction's numerator remains (š‘„+4)(š‘„+4).
The second fraction's numerator simplifies to -5(š‘„āˆ’4) = -5š‘„ + 20.

Step 5: Combine the fractions.
Now, we can combine the two fractions by keeping the LCD as the denominator:
(š‘„+4)(š‘„+4)/(š‘„+4)(š‘„āˆ’4) - 5š‘„ + 20/(š‘„+4)(š‘„āˆ’4).

Step 6: Simplify the numerator of the first fraction.
The numerator of the first fraction simplifies to (š‘„+4)(š‘„+4) = š‘„^2+8š‘„+16.

Step 7: Combine the fractions and simplify.
Now, we can combine the fractions by adding the numerators:
(š‘„^2+8š‘„+16)/(š‘„+4)(š‘„āˆ’4) - 5š‘„ + 20/(š‘„+4)(š‘„āˆ’4).

Finally, we simplify:
(š‘„^2+8š‘„+16 - 5š‘„ + 20)/(š‘„+4)(š‘„āˆ’4).

This simplifies to:
(š‘„^2+3š‘„+36)/(š‘„+4)(š‘„āˆ’4).