(4)/(a^2-4)-(1)/a^2+4a+4)=(1)/(a-2)

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To simplify this expression, we first need to factor the denominator in the second term on the left side:

(1)/(a^2+4a+4) = (1)/((a+2)(a+2)) = (1)/((a+2)^2)

Now we can rewrite the original expression:

(4)/(a^2-4)-(1)/((a+2)^2) = (1)/(a-2)

Next, we need to factor the denominator in the first term on the left side:

(a^2-4) = (a+2)(a-2)

Substitute these values back into the expression:

(4)/((a+2)(a-2))-(1)/((a+2)^2) = (1)/(a-2)

Now we can combine the fractions on the left side by finding a common denominator:

[4(a+2)] / [(a+2)(a-2)] - 1 / ((a+2)^2) = 1 / (a-2)

Now we can simplify further:

[4(a+2) - (a-2)] / [(a+2)(a-2)] = 1 / (a-2)

Solve the numerator:

[4a + 8 - a + 2] / [(a+2)(a-2)] = 1 / (a-2)

Combine like terms:

[3a + 10] / [(a+2)(a-2)] = 1 / (a-2)

Finally, cross multiply to solve for a:

(a-2)(3a + 10) = (a+2)(a-2)

Expand both sides:

3a^2 + 10a - 6a - 20 = a^2 - 4

Combine like terms:

3a^2 + 4a - 20 = a^2 - 4

Subtract a^2 from both sides:

2a^2 + 4a - 20 = -4

Rearrange so one side equals zero:

2a^2 + 4a - 16 = 0

Factor out a 2:

2(a^2 + 2a - 8) = 0

Factor the quadratic equation:

2(a+4)(a-2) = 0

Set each factor to zero:

a+4 = 0 or a-2 = 0

Therefore, a = -4 or a = 2.