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:

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