Factoring polynomials? Find x and know?

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HSG TEAM TEST TOPIC ANALYSIS OF RHTTT
SUBJECT: ENGLISH MATH 8 (TOPIC 2)
(Duration 90 minutes)

Lesson 1 (2.0 points). Factoring polynomials.
a) $4x^2 - y^2 + 4x + 1$
b) $x^5 - x^4 - 1$
c) $x(x + 4)(x + 10) + 128$
d) $(x - y)^2 + (y - z)^3 + (z - x)^3$

Lesson 2 (2.0 points). Find x and know.
a) $x^4 + 5x^3 - 8x - 40 = 0$
b) $9(5 - x) + x^2 - 10x = -25$

Lesson 3 (2.0 points).
a) Find the maximum value of the expression: $A = -4x^2 + x - 1$
b) Find the minimum value of the expression: $B = 2x^2 + 4x + 7$

Lesson 4 (2.0 points).
a) Prove that: $-4n^ - 4n^{2} + 16n$ is divisible by 384 for all even n greater than 4.
b) Find pairs of integers $(x, y)$ that satisfy the equality: $y^2 - 3x - 6 = 0$

Lesson 5 (2.0 points).
a) Given three numbers $a, b, c$ satisfying $a + b + c = 1$ and $a^2 + b^3 + c^3 = 1$. Prove that:
$a^{2015} + b^{2015} + c^{2015} = 1$
b) Let $a, b, c$ be positive numbers satisfying: $ab + bc + ac = 1$. Prove that:
$$\frac{a}{1 + a^2} + \frac{b}{1 + b^2} = \frac{1 + ab}{\sqrt{(1 + a^2)(1 + b^2)(1 + c^2)}}$$

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