Question

1. a.

4x
\

$$\begin{tabular}{|c|c|} \\hline $16x^2$ & \\\\ \\hline & 1 \\\\ \\hline \\end{tabular}$$

b.
\

$$\begin{tabular}{|c|c|} \\hline $10y$ & \\\\ \\hline & $-12x$ \\\\ \\hline \\end{tabular}$$

$2y$ $-3$

Explanation:

Step1: Find missing top-right (a)

We know the left factor is $4x$, and the bottom-right term is 1. First, find the top-right term by dividing $16x^2$ by $4x$ to get the top factor, then multiply by the right factor (which is $\frac{1}{4x} \times 1 = \frac{1}{4x}$? No, correct: the top row left is $4x \times 4x = 16x^2$, so the top factor is $4x$, right factor is $\frac{1}{4x} \times 1 = \frac{1}{4x}$? No, better: the right column factor is $\frac{1}{k}$ where $4x \times k =$ top-right, and $k \times$ bottom-left $=1$. First, bottom-left is $4x \times 1 = 4x$ (since left column is $4x$, bottom row right is 1, so bottom-left is $4x \times 1$). Then top-right is $4x \times \frac{1}{4x} \times 4x$? No, correct: the box is a multiplication table. Left side factors: $4x$ and $4x$? No, $4x \times 4x =16x^2$, so top factor is $4x$, right factor is $1$? No, bottom right is 1, so right factor is 1, bottom left factor is $1$. Wait, no: the left column is $4x$, so bottom left is $4x \times 1 =4x$, top right is $4x \times 1=4x$. Yes, that makes the full square: $(4x+1)^2 = 16x^2 +4x +4x +1$.
<Expression>Top-right (a): $4x \times 1 = 4x$</Expression>

Step2: Find missing bottom-left (a)

Bottom-left term is product of left factor $4x$ and bottom factor $1$.
<Expression>Bottom-left (a): $4x \times 1 = 4x$</Expression>

Step3: Find top-right (b)

First, find the top factor: $\frac{10y}{2y}=5$, so top factor is $5$. Top-right term is $5 \times (-3) = -15$.
<Expression>Top-right (b): $5 \times (-3) = -15$</Expression>

Step4: Find bottom-left (b)

Bottom-left term is product of $2y$ and the missing left factor? No, $\frac{-12x}{-3}=4x$, so bottom left is $2y \times 4x = 8xy$.
<Expression>Bottom-left (b): $2y \times 4x = 8xy$</Expression>

Answer:

a. Top-right: $\boldsymbol{4x}$, Bottom-left: $\boldsymbol{4x}$
b. Top-right: $\boldsymbol{-15}$, Bottom-left: $\boldsymbol{8xy}$