Via the FOIL method, we can attest that x(2x) + x(3) + –4(2x) + –4(3) = 2x² – 5x – 12.

Use FOIL: 

  (x+3)(x-5)(x) = (x² - 5x + 3x - 15)(x) = x³ - 5x² + 3x² - 15x = x³ - 2x² - 15x for A.

  (x-3)(x-1)(x+3) = (x-3)(x+3)(x-1) = (x² + 3x - 3x - 9)(x-1) = (x² - 9)(x-1)

  (x² - 9)(x-1) = x³ - x² - 9x _⁺ 9 for B. 

The difference between A and B: 

 (x³ - 2x² - 15x) - (x³ - x² - 9x _⁺ 9) = x³ - 2x² - 15x - x³ + x² + 9x - 9

 = - x² - 4x - 9. Since all of the terms are negative and x > 0:

  A - B < 0.

Rearrange A - B < 0:

  A < B

$\displaystyle x^3+5x^2-10=2x^2$

First, move all terms to one side of the equation to set them equal to zero.

$\displaystyle x^3+5x^2-2x^2-10x=0$

$\displaystyle x^3+3x^2-10x=0$

All terms contain an $\displaystyle \small x$, so we can factor it out of the equation.

$\displaystyle x(x^2+3x-10)=0$

Now, we can factor the quadratic in parenthesis. We need two numbers that add to $\displaystyle \small 3$ and multiply to $\displaystyle \small -10$.

$\displaystyle -2*5=-10\ \text{and}\ -2+5=3$

$\displaystyle x(x-2)(x+5)=0$

We now have three terms that multiply to equal zero. One of these terms must equal zero in order for the product to be zero.

$\displaystyle \begin{matrix} x=0 & x-2=0 &x+5=0 \\ x=0 & x=2 & x=-5 \end{matrix}$

Our answer will be $\displaystyle \small x=0,2,-5$.

This question can be solved using the FOIL method. So the first terms are multiplied together:

$\displaystyle (3x)(9x)$

This gives:

$\displaystyle 27x^2$

The x-squared is due to the x times x. 

The outer terms are then multipled together and added to the value above. 

$\displaystyle (3x)13=39x$

The inner two terms are multipled together to give the next term of the expression.

$\displaystyle (-4)(9x)=-36x$

Finally the last terms are multiplied together.

$\displaystyle (-4)(13)=-52$

All of the above terms are added together to give:

$\displaystyle 27x^2+39x-36x-52$

Combining like terms gives

$\displaystyle 27x^2+3x-52$.

Expand the following expression:

$\displaystyle (9x+6)(6x^2-3)$

Let's begin by recalling the meaning of FOIL: First, Outer, Inner, Last.

This means that in a situation such as we are given here, we need to multiply all the terms in a particular way. FOIL makes it easy to remember to multiply each pair of terms.

Let's begin:

First: 

$\displaystyle ({\color{Blue} 9x}+6)({\color{Blue} 6x^2}-3)\rightarrow9x*6x^2=54x^3$

Outer:

$\displaystyle ({\color{Blue} 9x}+6)(6x^2{\color{Blue} -3})\rightarrow9x*-3=-27x$

Inner:

$\displaystyle (9x+{\color{Blue} 6})({\color{Blue} 6x^2}-3)\rightarrow 6*6x^2=36x^2$

Last:

$\displaystyle (9x{\color{Blue} +6})(6x^2{\color{Blue} -3})=6*-3=-18$

Now, put it together in standard form to get:

$\displaystyle 54x^3+36x^2-27x-18$

$\displaystyle (x-3)^2=25$

Take the square root of both sides.

$\displaystyle x-3=\pm 5$

$\displaystyle x-3=5\ \text{or}\ x-3=-5$

Add 3 to both sides of each equation.

$\displaystyle x=5+3\ \text{or}\ x=-5+3$

$\displaystyle x=8\ \text{or}\ x=-2$

$\displaystyle \left ( xyz \right )^{3}+x^{2}y+\left ( xy \right )^{0}+\left ( xy^{\frac{1}{2}} \right )^{2}$

= x³y³z³ + x²y + x⁰y⁰ + x²y

= x³y³z³ + x²y + 1 + x²y

= x³y³z³ + 2x²y + 1

Use the FOIL method to simplify the following expression:

$\displaystyle (x^3+2x^2)^2$

Step 1: Expand the expression.

$\displaystyle (x^3+2x^2)(x^3+2x^2)$

Step 2: FOIL

First: $\displaystyle x^3\cdot x^3 = x^6$

Outside: $\displaystyle x^3 \cdot 2x^2 =2x^5$

Inside: $\displaystyle 2x^2 \cdot x^3 = 2x^5$

Last: $\displaystyle 2x^2 \cdot 2x^2 = 4x^4$

Step 2: Sum the products.

$\displaystyle x^6+2x^5+2x^5+4x^4$

$\displaystyle x^6+4x^5+4x^4$

$\displaystyle (x^{2}y^{4}+xy^{6})^{2}$

$\displaystyle (x^{2}y^{4}+xy^{6})(x^{2}y^{4}+xy^{6})$

We will need to FOIL.

First: $\displaystyle x^2y^4*x^2y^4=x^4y^8$

Inside: $\displaystyle xy^6*x^2y^4=x^3y^{10}$

Outside: $\displaystyle x^2y^4*xy^6=x^3y^{10}$

Last: $\displaystyle xy^6*xy^6=x^2y^{12}$

Sum all of the terms and simplify.

$\displaystyle x^4y^8+x^3y^{10}+x^3y^{10}+x^2y^{12}$

$\displaystyle x^4y^8+2x^3y^{10}+x^2y^{12}$

First calculate each section to yield 4c(27d³) – 8c³d + 2c⁴d⁴ = 108cd³ – 8c³d + 2c⁴d⁴. Now let's factor out the greatest common factor of the three terms, 2cd, in order to get:  2cd(54d² – 4c² + c³d³).