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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