$$
Group terms:
$$
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Question: Given $ h(x^2 + 1) = 2x^4 + 4x^2 + 3 $, find $ h(x^2 - 1) $.
9(x - 2)^2 - 36 - 4(y - 2)^2 + 16 = 44 9(x^2 - 4x) - 4(y^2 - 4y) = 44 $$

$$ 4m = 42 \Rightarrow m = \frac{42}{4} = \frac{21}{2} h(x^2 - 1) = 2(x^2 - 1)^2 + 1 = 2(x^4 - 2x^2 + 1) + 1 = 2x^4 - 4x^2 + 2 + 1 = 2x^4 - 4x^2 + 3 $$ 4m = 42 \Rightarrow m = \frac{42}{4} = \frac{21}{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 \frac{1}{2} \left( 1 + \frac{1}{2} - \frac{1}{51} - \frac{1}{52} \right) $$ $$
$$

Question: Compute $ \sum_{n=1}^{50} \frac{1}{n(n+2)} $.
$$ AreaQuestion: A microbiome researcher studying gut health models bacterial growth with the function $ f(x) = x^2 - 3x + m $, and models immune response with $ g(x) = x^2 - 3x + 3m $. If $ f(3) + g(3) = 42 $, what is the value of $ m $?
$$ So:

$$
$$

Question: Compute $ \sum_{n=1}^{50} \frac{1}{n(n+2)} $.
$$ AreaQuestion: A microbiome researcher studying gut health models bacterial growth with the function $ f(x) = x^2 - 3x + m $, and models immune response with $ g(x) = x^2 - 3x + 3m $. If $ f(3) + g(3) = 42 $, what is the value of $ m $?
$$ So:
\boxed{-2x - 2} Add the two expressions:
$$ Find common denominator for $ \frac{1}{51} + \frac{1}{52} $:
So the remainder is $ -2x - 2 $.


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

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AreaQuestion: A microbiome researcher studying gut health models bacterial growth with the function $ f(x) = x^2 - 3x + m $, and models immune response with $ g(x) = x^2 - 3x + 3m $. If $ f(3) + g(3) = 42 $, what is the value of $ m $?
$$ So:
\boxed{-2x - 2} Add the two expressions:
$$ Find common denominator for $ \frac{1}{51} + \frac{1}{52} $:
So the remainder is $ -2x - 2 $.

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

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$$ $$ Now substitute $ y = x^2 - 1 $:
a(\omega - \omega^2) = (\omega - \omega^2) + 3(\omega^2 - \omega) This is a telescoping series:
$$ $$
Now solve the system:
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Add the two expressions:
$$ Find common denominator for $ \frac{1}{51} + \frac{1}{52} $:
So the remainder is $ -2x - 2 $.

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

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$$ $$ Now substitute $ y = x^2 - 1 $:
a(\omega - \omega^2) = (\omega - \omega^2) + 3(\omega^2 - \omega) This is a telescoping series:
$$ $$
Now solve the system:
- Fourth: $ x - y = 4 $.
Now compute the sum:
The vertices are $ (4, 0), (0, 4), (-4, 0), (0, -4) $.
\frac{1}{2} \left( \frac{3}{2} - \frac{103}{2652} \right) = \frac{1}{2} \left( \frac{3978 - 103}{2652} \right) = \frac{1}{2} \cdot \frac{3875}{2652} = \frac{3875}{5304}

Question: Find the remainder when $ x^4 + 3x^2 + 1 $ is divided by $ x^2 + x + 1 $.
Solution: Use partial fractions to decompose the general term:
$$
Solution:
f(\omega) = \omega^4 + 3\omega^2 + 1 = \omega + 3\omega^2 + 1 = a\omega + b

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

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$$ $$ Now substitute $ y = x^2 - 1 $:
a(\omega - \omega^2) = (\omega - \omega^2) + 3(\omega^2 - \omega) This is a telescoping series:
$$ $$
Now solve the system:
- Fourth: $ x - y = 4 $.
Now compute the sum:
The vertices are $ (4, 0), (0, 4), (-4, 0), (0, -4) $.
\frac{1}{2} \left( \frac{3}{2} - \frac{103}{2652} \right) = \frac{1}{2} \left( \frac{3978 - 103}{2652} \right) = \frac{1}{2} \cdot \frac{3875}{2652} = \frac{3875}{5304}

Question: Find the remainder when $ x^4 + 3x^2 + 1 $ is divided by $ x^2 + x + 1 $.
Solution: Use partial fractions to decompose the general term:
$$
Solution:
f(\omega) = \omega^4 + 3\omega^2 + 1 = \omega + 3\omega^2 + 1 = a\omega + b $$ f(x) = (x^2 + x + 1)q(x) + ax + b Distribute and simplify:
Plug in $ x = \omega $:
$$
- Third: $ -x - y = 4 $, from $ (-4, 0) $ to $ (0, -4) $.
$$

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$$ 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 \