Solution: Perform polynomial long division or use the fact that the roots of $ x^2 + x + 1 = 0 $ are the non-real cube roots of unity, $ \omega $ and $ \omega^2 $, where $ \omega^3 = 1 $, $ \omega \ a(\omega - \omega^2) = (\omega - \omega^2) + 3(\omega^2 - \omega) Solution: Let $ y = x^2 + 1 \Rightarrow x^2 = y - 1 $.
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In the second: $ -x + y = 4 $, from $ (-4, 0) $ to $ (0, 4) $.
Let $ f(x) = x^4 + 3x^2 + 1 $. The remainder when dividing by a quadratic will be linear: $ ax + b $.
$$ $$ $$
9(x - 2)^2 - 4(y - 2)^2 = 60 a\omega^2 + b = \omega^2 + 3\omega + 1 \quad \ ext{(2)}
$$
9(x - 2)^2 - 4(y - 2)^2 = 60 a\omega^2 + b = \omega^2 + 3\omega + 1 \quad \ ext{(2)}
$$

Question: Compute $ \sum_{n=1}^{50} \frac{1}{n(n+2)} $.
9(x^2 - 4x) - 4(y^2 - 4y) = 44 $$


e 1 $, and $ \omega^2 + \omega + 1 = 0 $.
$$

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9(x^2 - 4x) - 4(y^2 - 4y) = 44 $$

e 1 $, and $ \omega^2 + \omega + 1 = 0 $.
$$

Why are travelers increasingly talking about Miami International Airport’s grab-and-go car rental spot? Known for its convenient location and efficient transfers, this often-overlooked airport car rental hub is quietly becoming a smart choice for travelers seeking speed, simplicity, and savings. Now hailed as the ultimate hidden gem, Hit the Road at Miami Airport delivers seamless mobility solutions that cut through the chaos of traditional car rental lines.

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$$
Substitute into the expression:
$$ $$
$$
$$
$$ This is a telescoping series:
$$

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$$
Substitute into the expression:
$$ $$
$$
$$
$$ This is a telescoping series:
Now substitute $ y = x^2 - 1 $:
$$
$$
Solution:
f(\omega) = \omega^4 + 3\omega^2 + 1 = \omega + 3\omega^2 + 1 = a\omega + b Find common denominator for $ \frac{1}{51} + \frac{1}{52} $:
$$

Question: Find the remainder when $ x^4 + 3x^2 + 1 $ is divided by $ x^2 + x + 1 $.

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Substitute into the expression:
$$ $$
$$
$$
$$ This is a telescoping series:
Now substitute $ y = x^2 - 1 $:
$$
$$
Solution:
f(\omega) = \omega^4 + 3\omega^2 + 1 = \omega + 3\omega^2 + 1 = a\omega + b Find common denominator for $ \frac{1}{51} + \frac{1}{52} $:
$$

Question: Find the remainder when $ x^4 + 3x^2 + 1 $ is divided by $ x^2 + x + 1 $.
$$ Add the two expressions:
\boxed{(2, 2)} h(x^2 - 1) = 2(x^2 - 1)^2 + 1 = 2(x^4 - 2x^2 + 1) + 1 = 2x^4 - 4x^2 + 2 + 1 = 2x^4 - 4x^2 + 3 Now compute the sum:
$$ Subtract (1) - (2):
f(3) + g(3) = m + 3m = 4m This is a hyperbola centered at $ (2, 2) $.

$$
$$ This is a telescoping series:
Now substitute $ y = x^2 - 1 $:
$$
$$
Solution:
f(\omega) = \omega^4 + 3\omega^2 + 1 = \omega + 3\omega^2 + 1 = a\omega + b Find common denominator for $ \frac{1}{51} + \frac{1}{52} $:
$$

Question: Find the remainder when $ x^4 + 3x^2 + 1 $ is divided by $ x^2 + x + 1 $.
$$ Add the two expressions:
\boxed{(2, 2)} h(x^2 - 1) = 2(x^2 - 1)^2 + 1 = 2(x^4 - 2x^2 + 1) + 1 = 2x^4 - 4x^2 + 2 + 1 = 2x^4 - 4x^2 + 3 Now compute the sum:
$$ Subtract (1) - (2):
f(3) + g(3) = m + 3m = 4m This is a hyperbola centered at $ (2, 2) $.
$$ Evaluate $ g(3) $:
Group terms:
$$


$$ $$ $$ \boxed{\frac{3875}{5304}} $$