1. Find the products. a) $$ (x+2)(x+3) $$ c) $$ (a-2)(a+3) $$ e) $$ (2 y+5)(3 y-2) $$ g) $$ (2 a-3)(3 a-2) $$ i) $$ (x-3)(x-2) $$ k) $$ (2 x-3 y)(2 x+4 y) $$ m) $$ (7 x+3 y)(2 x-4 y) $$

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UpStudy AI · Accepted Answer

Sure! Let's find the products for each of the given binomials by using the distributive property (FOIL method).

### a) $$ (x + 2)(x + 3) $$

$$
\begin{align*}
(x + 2)(x + 3) &= x \cdot x + x \cdot 3 + 2 \cdot x + 2 \cdot 3 \
&= x^2 + 3x + 2x + 6 \
&= x^2 + 5x + 6
\end{align*}
$$

**Product:** $$ x^2 + 5x + 6 $$

---

### c) $$ (a - 2)(a + 3) $$

$$
\begin{align*}
(a - 2)(a + 3) &= a \cdot a + a \cdot 3 - 2 \cdot a - 2 \cdot 3 \
&= a^2 + 3a - 2a - 6 \
&= a^2 + a - 6
\end{align*}
$$

**Product:** $$ a^2 + a - 6 $$

---

### e) $$ (2y + 5)(3y - 2) $$

$$
\begin{align*}
(2y + 5)(3y - 2) &= 2y \cdot 3y + 2y \cdot (-2) + 5 \cdot 3y + 5 \cdot (-2) \
&= 6y^2 - 4y + 15y - 10 \
&= 6y^2 + 11y - 10
\end{align*}
$$

**Product:** $$ 6y^2 + 11y - 10 $$

---

### g) $$ (2a - 3)(3a - 2) $$

$$
\begin{align*}
(2a - 3)(3a - 2) &= 2a \cdot 3a + 2a \cdot (-2) - 3 \cdot 3a - 3 \cdot (-2) \
&= 6a^2 - 4a - 9a + 6 \
&= 6a^2 - 13a + 6
\end{align*}
$$

**Product:** $$ 6a^2 - 13a + 6 $$

---

### i) $$ (x - 3)(x - 2) $$

$$
\begin{align*}
(x - 3)(x - 2) &= x \cdot x + x \cdot (-2) - 3 \cdot x - 3 \cdot (-2) \
&= x^2 - 2x - 3x + 6 \
&= x^2 - 5x + 6
\end{align*}
$$

**Product:** $$ x^2 - 5x + 6 $$

---

### k) $$ (2x - 3y)(2x + 4y) $$

$$
\begin{align*}
(2x - 3y)(2x + 4y) &= 2x \cdot 2x + 2x \cdot 4y - 3y \cdot 2x - 3y \cdot 4y \
&= 4x^2 + 8xy - 6xy - 12y^2 \
&= 4x^2 + 2xy - 12y^2
\end{align*}
$$

**Product:** $$ 4x^2 + 2xy - 12y^2 $$

---

### m) $$ (7x + 3y)(2x - 4y) $$

$$
\begin{align*}
(7x + 3y)(2x - 4y) &= 7x \cdot 2x + 7x \cdot (-4y) + 3y \cdot 2x + 3y \cdot (-4y) \
&= 14x^2 - 28xy + 6xy - 12y^2 \
&= 14x^2 - 22xy - 12y^2
\end{align*}
$$

**Product:** $$ 14x^2 - 22xy - 12y^2 $$

---

Feel free to ask if you need further explanations or assistance with other problems!
Here are the products of the given binomials: a) $$ (x + 2)(x + 3) = x^2 + 5x + 6 $$ c) $$ (a - 2)(a + 3) = a^2 + a - 6 $$ e) $$ (2y + 5)(3y - 2) = 6y^2 + 11y - 10 $$ g) $$ (2a - 3)(3a - 2) = 6a^2 - 13a + 6 $$ i) $$ (x - 3)(x - 2) = x^2 - 5x + 6 $$ k) $$ (2x - 3y)(2x + 4y) = 4x^2 + 2xy - 12y^2 $$ m) $$ (7x + 3y)(2x - 4y) = 14x^2 - 22xy - 12y^2 $$

1. Find the products. a) \( (x+2)(x+3) \) c) \( (a-2)(a+3) \) e) \( (2 y+5)(3 y-2) \) g) \( (2 a-3)(3 a-2) \) i) \( (x-3)(x-2) \) k) \( (2 x-3 y)(2 x+4 y) \) m) \( (7 x+3 y)(2 x-4 y) \)

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