Section 2.5 - Multiplying A Polynomial By A Monomial

Explanation
The given expression is 2(-c^2 + 5c - 3). To expand this expression, we distribute the 2 to each term inside the parentheses. This gives us -2c^2 + 10c - 6. Therefore, the correct answer is -2c^2 + 10c - 6.

Explanation
The given expression can be expanded by distributing -4y^2 to each term inside the parentheses. This results in -8y^4 + 20y^3 - 24y^2.

Explanation
The given expression is -w(5w^2 + w - 6). To expand this expression, we distribute the -w to each term inside the parentheses. This gives us -5w^3 - w^2 + 6w. Therefore, the correct answer is -5w^3 - w^2 + 6w.

Explanation
To find the missing factor, we need to solve the equation. We can start by distributing the factor to the terms inside the parentheses: -2(4x - 7) = -8x + 14. This simplifies to -8x + 14 = -8x + 14. Since the variables and constants are the same on both sides of the equation, this means that any value for the missing factor would satisfy the equation. Therefore, the missing factor could be any real number, including -2.

Explanation
The given equation is -3m(______) = 12m^3 - 9m^2 + 6m. To find the missing terms, we need to divide both sides of the equation by -3m. This will give us the missing terms on the left side. Dividing each term on the right side by -3m, we get -4m^2 -3m -2. Therefore, the missing terms in the equation are -4m^2 -3m -2.

Explanation
In order to find the missing terms, we can compare the given equation with the answer choices. We notice that the first term in the equation is 2xy, which matches with the term 3xy in the answer choice. The second term in the equation is 6x^2y, which matches with the term 3x in the answer choice. Finally, the third term in the equation is -2xy^3, which matches with the term -y^2 in the answer choice. Therefore, the missing terms in 2xy(_________) = 6x^2y - 2xy^3 + 6x^2y^2 are 3x - y^2 + 3xy.

Explanation
The given expression represents the area of the figure shown above. It is obtained by multiplying the length and width of the figure, which are represented by the terms 30z^2 and -15z respectively. Therefore, the simplified algebraic expression for the area is 30z^2 - 15z.

Explanation
The given answer, 25n^2 + 10n, represents a simplified algebraic expression for the area of the parallelogram. It correctly combines the base and height of the parallelogram, represented by the variables n and 5n, respectively. By multiplying these two terms, we get 25n^2. Additionally, the expression includes the area of the rectangle on top of the parallelogram, which is represented by the term 10n. Therefore, the expression 25n^2 + 10n is the correct simplified algebraic expression for the area of the figure.

Explanation
The given expression represents the area of a trapezoid. The formula for the area of a trapezoid is [(top + bottom)(height)] / 2. In this case, the expression 36x^2 + 9x represents the area of the trapezoid. The term 36x^2 represents the product of the top and bottom lengths of the trapezoid, and the term 9x represents the height of the trapezoid. Therefore, the expression 36x^2 + 9x is the simplified algebraic expression for the area of the trapezoid.