Here we have to use the distributive property and FOIL when we are multiplying the two terms. First we multipy the terms in the binomial with the trinomial. Then we add or subtract the coefficients of the terms with same combination of variables x and y.

\(\displaystyle (x+3y) (y^2+3xy+2)\)

\(\displaystyle = x(y^2+3xy+2) +3y(y^2+3xy+2)\)

\(\displaystyle = xy^2+3x^2y+2x +3y^3+9xy^2+6y\)

\(\displaystyle = 3y^3+10xy^2+3x^2y+2x+6y\)

Use the FOIL method to multiply the terms.

\(\displaystyle (3x-6)(3-x) = (3x)(3) +(3x)(-x)+(-6)(3)+(-6)(-x)\)

Simplify the terms on the right side of the equation.

\(\displaystyle 9x-3x^2-18+6x\)

Combine like-terms.

The answer is:  \(\displaystyle -3x^2+15x-18\)

As we know, FOIL stands for First, Outside, Inside, Last, which indicates the order in which we should multiply each of these terms.

First: \(\displaystyle 4x\cdot2x=8x^{2}\)

Outside: \(\displaystyle 4x\cdot-6=-24x\)

Inside: \(\displaystyle 3\cdot2x=6x\)

Last: \(\displaystyle 3\cdot-6=-18\)

We then combine like terms, which in this case are \(\displaystyle -24x\) and \(\displaystyle 6x\). When we add these together, we are left with a trinomial of \(\displaystyle 8x^{2}-18x-18\). None of the remaining terms are alike, so there is no further simplification possible.

To expand this set of binomials we need to FOIL the terms. FOIL is the multiplication steps that need to be applied to a set of binomials.

\(\displaystyle (5-4x)(-3+x)\)

First:  \(\displaystyle 5\cdot -3=-15\)

Outer:  \(\displaystyle 5\cdot x=5x\)

Inner:\(\displaystyle -4x\cdot-3=12x\)

Last:\(\displaystyle -4x\cdotx=-4x^{2}\)

\(\displaystyle $Sum$ =-15+5x+12x-4x^2 = -4x^2+17x-15\)

Use the FOIL method to multiply the polynomials:

(3x+4)(x-2)

 F - Multiply the first terms of each binomial

\(\displaystyle 3x * x = 3x^2\)

O- Multiply the outside terms of each binomial

\(\displaystyle 3x*-2 = -6x\)

I- Multiply the inside terms of each binomial

 \(\displaystyle 4*x=4x\)

L- Multiply the last terms of each binomial

\(\displaystyle 4*-2=-8\)

Add all of the answers and combine like terms

\(\displaystyle 3x^2-6x+4x-8\)

\(\displaystyle 3x^2-2x-8\)

Use FOIL to distribute each term.

F: \(\displaystyle 3x(4y)=12xy\)

O: \(\displaystyle 3x(7)=21x\)

I: \(\displaystyle -8(4y)=-32y\)

L:\(\displaystyle -8(7)=-56\)

If possible, combine like terms (none in this case).

\(\displaystyle 12xy+21x-32y-56\)

For an expression in the form \(\displaystyle ax^{2}+bx+c\), in its simplest form where \(\displaystyle a=1\), find two integers whose sum is \(\displaystyle b\) and whose product is \(\displaystyle c\).

In the case of

3 and 4 fit the requirements. Therefore, our answer looks like:

For an expression in the form \(\displaystyle ax^{2}+bx+c\), in its simplest form where \(\displaystyle a=1\), find two integers whose sum is \(\displaystyle b\) and whose product is \(\displaystyle c\).

In the case of

\(\displaystyle x^{2}-2x-63\)

\(\displaystyle -9\) and \(\displaystyle 7\) fit the requirements. Therefore, our answer looks like: