Extended Distributive Property: Using variables

Let's simplify the given expression, open the parentheses using the extended distribution law:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

Note that in the formula template for the above distribution law, we take by default that the operation between the terms inside the parentheses is addition, therefore we won't forget of course that the sign preceding the term is an inseparable part of it, we will also apply the rules of sign multiplication and thus we can present any expression in parentheses, which we'll open using the above formula, first as an expression where addition operation exists between all terms, in this expression as clear to all the terms' preceding sign is - plus, therefore we'll proceed directly to opening the parentheses,

$$Let's begin then with opening the parentheses:

$(\textcolor{red}{7x}+\textcolor{blue}{4})(3x+4)\\ \textcolor{red}{7x}\cdot3x+ \textcolor{red}{7x}\cdot4+\textcolor{blue}{4}\cdot 3x +\textcolor{blue}{4}\cdot4\\ 21x^2+28x+12x+16$

In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

In the next step we'll combine like terms, we'll define like terms as terms where the variable (or variables each separately), in this case x, have identical exponents (in the absence of one of the variables from the expression, we'll treat its exponent as zero power since raising any number to the zero power yields the result 1), we'll use the commutative property of addition, additionally we'll arrange (if needed) the expression from highest to lowest power from left to right (we'll treat the free number as zero power):
$\textcolor{purple}{21x^2}\textcolor{green}{+28x}\textcolor{green}{+12x}+16\\ \textcolor{purple}{21x^2}\textcolor{green}{+40x}+16$

In the combining of like terms performed above, we highlighted the different terms using colors, and as emphasized before, we made sure that the sign preceding the term is an inseparable part of it,

We therefore got that the correct answer is answer B.

We begin by solving the addition exercise in the right parenthesis:

$(7x+3)+14=238$

We then multiply each of the terms inside of the parentheses by 14:

$(14\times7x)+(14\times3)=238$

Following this we solve each of the exercises inside of the parentheses:

$98x+42=238$

We move the sections whilst retaining the appropriate sign:

$98x=238-42$

$98x=196$

Finally we divide the two parts by 98:

$\frac{98}{98}x=\frac{196}{98}$

$x=2$

We begin by solving the addition exercise in the right parenthesis:

$(9+17x)\times7=420$

We then multiply each of the terms inside the parentheses by 7:

$(9\times7)+(17x\times7)=420$

We continue by solving each of the exercises inside of the parentheses:

$63+119x=420$

Following this we rearrange the sections whilst maintaining the appropriate sign:

$119x=420-63$

$119x=357$

Finally we divide the two parts by 119:

$\frac{119}{119}x=\frac{357}{119}$

$x=3$

We begin by solving the two exercises inside of the parentheses:

$4a\times7=112$

We then divide each of the sections by 4:

$\frac{4a\times7}{4}=\frac{112}{4}$

In the fraction on the left side we simplify by 4 and in the fraction on the right side we divide by 4:

$a\times7=28$

Remember that:

$a\times7=a7$

Lastly we divide both sections by 7:

$\frac{a7}{7}=\frac{28}{7}$

$a=4$

We know that the area of a rectangle is equal to its length multiplied by its width.

We begin by writing an equation with the available data.

$(4x+x^2)\times(3x+8+5x)$

Next we use the distributive property to solve the equation.

$(4x\times3x)+(4x\times8)+(4x\times5x)+(x^2\times3x)+(x^2\times8)+(x^2\times5x)=$

We then solve each of the exercises within the parentheses:

$12x^2+32x+20x^2+3x^3+16x^2+5x^3=$

Finally we add up all the coefficients of X squared and all the coefficients of X cubed and we obtain the following:

$48x^2+8x^3+32x$