The document you uploaded appears to be a math exercise sheet in Russian. Here's a breakdown of the tasks based on the image:

1. Представьте на множители: Factorize the expressions.
2. Разложите на множители: Factorize the expression.
3. Упростите выражение: Simplify the expression.
4. Решите уравнение: Solve the equation.
5. Представьте выражение в виде произведения: Represent the expression as a product.
6. Упростите выражение: Simplify the expression again.
7. Докажите, что выражение: Prove that the expression holds for all values of $x$.

I will address each task in sequence:

Task 1:

Factorize the expression $(x + 8)^2$.

This is a perfect square, and the factorization is straightforward: $(x + 8)^2 = (x + 8)(x + 8)$

Task 2:

Factorize the expression $(16 - c^2)$.

This is a difference of squares. It can be factored as: $16 - c^2 = (4 - c)(4 + c)$

Task 3:

Simplify the expression $(10x - 3)^2$.

Expanding the square: $(10x - 3)^2 = (10x)^2 - 2(10x)(3) + 3^2 = 100x^2 - 60x + 9$

Task 4:

Solve the equation $(2x - 7)(x + 1) = (3x + 1)(4x - 5)$.

Expanding both sides:

• Left side: $(2x - 7)(x + 1) = 2x^2 + 2x - 7x - 7 = 2x^2 - 5x - 7$
• Right side: $(3x + 1)(4x - 5) = 12x^2 - 15x + 4x - 5 = 12x^2 - 11x - 5$

Equating both sides: $2x^2 - 5x - 7 = 12x^2 - 11x - 5$ Bringing all terms to one side: $2x^2 - 5x - 7 - 12x^2 + 11x + 5 = 0$ Simplifying: $-10x^2 + 6x - 2 = 0$ Dividing the equation by -2: $5x^2 - 3x + 1 = 0$ Solving this quadratic equation using the quadratic formula: $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(5)(1)}}{2(5)} = \frac{3 \pm \sqrt{9 - 20}}{10} = \frac{3 \pm \sqrt{-11}}{10}$ Since the discriminant is negative ($\sqrt{-11}$), there are no real solutions.

Task 5:

Represent the expression $(2 - x)(x^2/2) + (x^2 - 2x + 5)(x + 5)$ as a product.

Let's simplify and factor this expression:

• $(2 - x)(x^2/2) = x^2 - 2x$
• $(x^2 - 2x + 5)(x + 5)$ can be expanded, but factoring it directly looks complex. However, we can see that this expression does not factor neatly into a simple product. We can either expand it completely or look for a common factor in the components.

Task 6:

Simplify the expression $(x - 9)(x + 9)$.

This is a difference of squares: $(x - 9)(x + 9) = x^2 - 81$

Task 7:

Prove that the expression $x^2 - 18x + 84$ takes positive values for all values of $x$.

We can complete the square for this quadratic expression: $x^2 - 18x + 84 = (x - 9)^2 + 3$ Since $(x - 9)^2$ is always non-negative and 3 is positive, the entire expression is always greater than or equal to 3, meaning it is always positive.

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Would you like further details or any clarifications on these solutions?

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Here are some follow-up questions you might find interesting:

1. How do you solve quadratic equations using the quadratic formula?
2. Can you explain more about the difference of squares?
3. What are other methods for factoring expressions besides the difference of squares?
4. How do you complete the square for quadratic expressions?
5. What other types of proofs can be done with quadratic expressions?

Tip: When factoring quadratics, always check if you can apply the difference of squares, perfect square trinomials, or other known formulas to simplify the process.