1. Find the products. $$ \begin{array}{ll} ext { a) }(x+2)(x+3) & ext { (b) }(p-3)(p-4) \ ext { c) }(a-2)(a+3) & ext { d) }(2 x-3)(3 x+1) \ ext { e) }(2 y+5)(3 y-2) & ext { f) }(3 a-7)(2 a+1) \ ext { g) }(2 a-3)(3 a-2) & ext { (h) }(4 y+1)(7 y-2) \ ext { i) }(x-3)(x-2) & ext { j) }(4 a-2)(3 a+5) \ ext { k) }(2 x-3 y)(2 x+4 y) & ext { 1) }(2 a-7 b)(3 a-2 b) \ ext { m) }(7 x+3 y)(2 x-4 y) & ext { n) }(3 a-2 b)(2 a+11 b)\end{array} $$

Question

1. Find the products. $$ \begin{array}{ll}	ext { a) }(x+2)(x+3) & 	ext { (b) }(p-3)(p-4) \ 	ext { c) }(a-2)(a+3) & 	ext { d) }(2 x-3)(3 x+1) \ 	ext { e) }(2 y+5)(3 y-2) & 	ext { f) }(3 a-7)(2 a+1) \ 	ext { g) }(2 a-3)(3 a-2) & 	ext { (h) }(4 y+1)(7 y-2) \ 	ext { i) }(x-3)(x-2) & 	ext { j) }(4 a-2)(3 a+5) \ 	ext { k) }(2 x-3 y)(2 x+4 y) & 	ext { 1) }(2 a-7 b)(3 a-2 b) \ 	ext { m) }(7 x+3 y)(2 x-4 y) & 	ext { n) }(3 a-2 b)(2 a+11 b)\end{array} $$

UpStudy AI · Accepted Answer

Certainly! Let's expand each of the given binomial products by applying the distributive property (also known as the FOIL method for binomials):

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

### b) $$ (p - 3)(p - 4) $$
$$
\begin{align*}
(p - 3)(p - 4) &= p \cdot p + p \cdot (-4) + (-3) \cdot p + (-3) \cdot (-4) \
&= p^2 - 4p - 3p + 12 \
&= p^2 - 7p + 12
\end{align*}
$$

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

### d) $$ (2x - 3)(3x + 1) $$
$$
\begin{align*}
(2x - 3)(3x + 1) &= 2x \cdot 3x + 2x \cdot 1 + (-3) \cdot 3x + (-3) \cdot 1 \
&= 6x^2 + 2x - 9x - 3 \
&= 6x^2 - 7x - 3
\end{align*}
$$

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

### f) $$ (3a - 7)(2a + 1) $$
$$
\begin{align*}
(3a - 7)(2a + 1) &= 3a \cdot 2a + 3a \cdot 1 + (-7) \cdot 2a + (-7) \cdot 1 \
&= 6a^2 + 3a - 14a - 7 \
&= 6a^2 - 11a - 7
\end{align*}
$$

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

### h) $$ (4y + 1)(7y - 2) $$
$$
\begin{align*}
(4y + 1)(7y - 2) &= 4y \cdot 7y + 4y \cdot (-2) + 1 \cdot 7y + 1 \cdot (-2) \
&= 28y^2 - 8y + 7y - 2 \
&= 28y^2 - y - 2
\end{align*}
$$

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

### j) $$ (4a - 2)(3a + 5) $$
$$
\begin{align*}
(4a - 2)(3a + 5) &= 4a \cdot 3a + 4a \cdot 5 + (-2) \cdot 3a + (-2) \cdot 5 \
&= 12a^2 + 20a - 6a - 10 \
&= 12a^2 + 14a - 10
\end{align*}
$$

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

### l) $$ (2a - 7b)(3a - 2b) $$
$$
\begin{align*}
(2a - 7b)(3a - 2b) &= 2a \cdot 3a + 2a \cdot (-2b) + (-7b) \cdot 3a + (-7b) \cdot (-2b) \
&= 6a^2 - 4ab - 21ab + 14b^2 \
&= 6a^2 - 25ab + 14b^2
\end{align*}
$$

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

### n) $$ (3a - 2b)(2a + 11b) $$
$$
\begin{align*}
(3a - 2b)(2a + 11b) &= 3a \cdot 2a + 3a \cdot 11b + (-2b) \cdot 2a + (-2b) \cdot 11b \
&= 6a^2 + 33ab - 4ab - 22b^2 \
&= 6a^2 + 29ab - 22b^2
\end{align*}
$$

---

**Summary of Results:**

- **a)** $$ x^2 + 5x + 6 $$
- **b)** $$ p^2 - 7p + 12 $$
- **c)** $$ a^2 + a - 6 $$
- **d)** $$ 6x^2 - 7x - 3 $$
- **e)** $$ 6y^2 + 11y - 10 $$
- **f)** $$ 6a^2 - 11a - 7 $$
- **g)** $$ 6a^2 - 13a + 6 $$
- **h)** $$ 28y^2 - y - 2 $$
- **i)** $$ x^2 - 5x + 6 $$
- **j)** $$ 12a^2 + 14a - 10 $$
- **k)** $$ 4x^2 + 2xy - 12y^2 $$
- **l)** $$ 6a^2 - 25ab + 14b^2 $$
- **m)** $$ 14x^2 - 22xy - 12y^2 $$
- **n)** $$ 6a^2 + 29ab - 22b^2 $$
Here are the expanded forms of the given binomial products: a) $$ x^2 + 5x + 6 $$ b) $$ p^2 - 7p + 12 $$ c) $$ a^2 + a - 6 $$ d) $$ 6x^2 - 7x - 3 $$ e) $$ 6y^2 + 11y - 10 $$ f) $$ 6a^2 - 11a - 7 $$ g) $$ 6a^2 - 13a + 6 $$ h) $$ 28y^2 - y - 2 $$ i) $$ x^2 - 5x + 6 $$ j) $$ 12a^2 + 14a - 10 $$ k) $$ 4x^2 + 2xy - 12y^2 $$ l) $$ 6a^2 - 25ab + 14b^2 $$ m) $$ 14x^2 - 22xy - 12y^2 $$ n) $$ 6a^2 + 29ab - 22b^2 $$

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