First distribute the \(\displaystyle -4x\) to all terms in the parenthesis.

\(\displaystyle (-4x \times -2x^{2}) + (-4x \times 3y) + (-4x \times -4)\)

\(\displaystyle (-4x \times -2x^{2}) = 8x ^{1+2} = 8x^{3}\)  (When multiplying variables with exponents, the exponents are added together).

\(\displaystyle (-4x \times 3y) = -4\times 3 \times x\times y = -12xy\)

\(\displaystyle -4x (-4) = 16x\)

\(\displaystyle 8x^{3} -12xy + 16x\)

To solve \(\displaystyle (-6x^{4}y^{2}) \times(-3x^{2}y^{2})\)

Separate the parts of the terms.  Multiply the coefficients:

\(\displaystyle -6 \times - 3 = 18\)

Multiply the terms with \(\displaystyle x\) as the variable. When multiplying, the exponents get added.

\(\displaystyle x^{4}\times x^{2} = x^{4+2} = x^{6}\)

Multiply the terms with \(\displaystyle y\) as the variable. When multiplying, the exponents get added.

\(\displaystyle y^{2}\times y^{2} = y^{2+2} = y^{4}\)

Put it all together:

\(\displaystyle 18x^{6}y^{4}\)

The formula for the Area of a rectangle is length times width or l x w.  Multiply the length which is \(\displaystyle 4y\) times the width which is \(\displaystyle 9y.\)

First multiply

\(\displaystyle 4 \times 9 =36\)

Then multiply the variables; add the exponents:

\(\displaystyle y\timesy = y^{1+1} = y^{2}\)

Area or A of the rectangle is

\(\displaystyle 36y^{2}\)

The volume of a rectangular prism is determined by using the formula

V = length x width x height

\(\displaystyle 8 \times 6\times 5 = 240\)

\(\displaystyle x \ast x \ast x = x^{1+1+1} =x^{3}\)

\(\displaystyle 240x^{3} centimeters^{3}\)

When multiplying variables, we multiply them together just like we would integers.  We have

\(\displaystyle x \cdot 3x\)

The first term isn't showing a coefficient, so it is automatically 1.  So,

\(\displaystyle 1x \cdot 3x\)

Now, we will multiply the coefficients together, then we multiply the variables together.

So, we get

\(\displaystyle 1x \cdot 3x = (1 \cdot 3)(x \cdot x) = 3x^2\)

Simplify the following expression:

\(\displaystyle 9x^3t^7*4x^5t^{12}\)

Let's begin by rewriting the expression with similar terms next to each other.

\(\displaystyle (9*4)(x^3*x^5)(t^7*t^{12})\)

Next, multiply the terms out.

Our first pair of terms is just like regular multiplication.

For the next two terms, we need to add the exponents. 

In doing so, we get:

\(\displaystyle 36x^8t^{19}\)

Simplify the following statement

\(\displaystyle (14b^2t^5)(2b^3t^9)\)

To multiply these terms, first multiply the 2 and the 14

\(\displaystyle 28(b^2t^5)(b^3t^9)\)

Next, we can combine the b's and the t's by adding together their exponents.

\(\displaystyle 28(b^2t^5)(b^3t^9)=28b^5t^{14}\)

So our answer is:

\(\displaystyle 28b^5t^{14}\)

Combine the following terms:

\(\displaystyle 5q^3*8q^4*9q^{11}\)

Let's begin by multiplying the coefficients (numbers out in front)

\(\displaystyle 5*8*9=40*9=360\)

So our answer must have 360 in front.

Next, let's multiply our q's. To do so, we need to add up our exponents.

\(\displaystyle q^3*q^4*q^{11}=q^{3+4+11}=q^{18}\)

So, put it together to get:

\(\displaystyle 360q^{18}\)

\(\displaystyle x^{2} = 64\), so either \(\displaystyle x = \sqrt{64} = 8\) or \(\displaystyle x = - \sqrt{64} =- 8\).

If the former is true, 

\(\displaystyle x^{4} = 8^{4} = 8 \cdot 8 \cdot 8 \cdot 8 = 4,096\).

If the latter is true, then, since an even-numbered power of any number is positive, 

\(\displaystyle x^{4} =( - 8)^{4} = 8^{4} = 4,096\).

Simplify the following expression

\(\displaystyle 54x^3*8x^2*\frac{x^2}{9}\)

Let's begin by grouping our x's and our coefficients

\(\displaystyle (54*8*\frac{1}{9})(x^3*x^2*x^2)\)

We can combine our coefficents by multiplying them, and we can combine the x's by adding the exponents

\(\displaystyle (54*\frac{1}{9}*8)(x^7)=(6*8)x^7=48x^7\)

So, our answer is:

\(\displaystyle 48x^7\)