Question

a) (3s + 5)(2s + 2) + (3s + 7)(s + 6)
b) (2x + 3)(5x + 4) + (x - 4)(3x - 7)

Explanation:

Let's solve part (a) and part (b) one by one.

Part (a)

We need to expand and simplify \((3s + 5)(2s + 2)+(3s + 7)(s + 6)\)

Step 1: Expand each product using the distributive property (FOIL method)

• Expand \((3s + 5)(2s + 2)\):

\[

$$\begin{align*} (3s + 5)(2s + 2)&=3s\times2s+3s\times2 + 5\times2s+5\times2\\ &=6s^{2}+6s + 10s+10\\ &=6s^{2}+16s + 10 \end{align*}$$

\]

• Expand \((3s + 7)(s + 6)\):

\[

$$\begin{align*} (3s + 7)(s + 6)&=3s\times s+3s\times6+7\times s + 7\times6\\ &=3s^{2}+18s+7s + 42\\ &=3s^{2}+25s + 42 \end{align*}$$

\]

Step 2: Add the two expanded expressions together

\[

$$\begin{align*} &(6s^{2}+16s + 10)+(3s^{2}+25s + 42)\\ &=6s^{2}+3s^{2}+16s+25s + 10 + 42\\ &=9s^{2}+41s+52 \end{align*}$$

\]

Part (b)

We need to expand and simplify \((2x + 3)(5x + 4)+(x - 4)(3x - 7)\)

Step 1: Expand each product using the distributive property (FOIL method)

• Expand \((2x + 3)(5x + 4)\):

\[

$$\begin{align*} (2x + 3)(5x + 4)&=2x\times5x+2x\times4+3\times5x + 3\times4\\ &=10x^{2}+8x+15x + 12\\ &=10x^{2}+23x + 12 \end{align*}$$

\]

• Expand \((x - 4)(3x - 7)\):

\[

$$\begin{align*} (x - 4)(3x - 7)&=x\times3x-x\times7-4\times3x+4\times7\\ &=3x^{2}-7x-12x + 28\\ &=3x^{2}-19x + 28 \end{align*}$$

\]

Step 2: Add the two expanded expressions together

\[

$$\begin{align*} &(10x^{2}+23x + 12)+(3x^{2}-19x + 28)\\ &=10x^{2}+3x^{2}+23x-19x+12 + 28\\ &=13x^{2}+4x + 40 \end{align*}$$

\]

Final Answers

a) \(\boldsymbol{9s^{2}+41s + 52}\)

b) \(\boldsymbol{13x^{2}+4x + 40}\)

Answer:

Let's solve part (a) and part (b) one by one.

Part (a)

We need to expand and simplify \((3s + 5)(2s + 2)+(3s + 7)(s + 6)\)

Step 1: Expand each product using the distributive property (FOIL method)

• Expand \((3s + 5)(2s + 2)\):

\[

$$\begin{align*} (3s + 5)(2s + 2)&=3s\times2s+3s\times2 + 5\times2s+5\times2\\ &=6s^{2}+6s + 10s+10\\ &=6s^{2}+16s + 10 \end{align*}$$

\]

• Expand \((3s + 7)(s + 6)\):

\[

$$\begin{align*} (3s + 7)(s + 6)&=3s\times s+3s\times6+7\times s + 7\times6\\ &=3s^{2}+18s+7s + 42\\ &=3s^{2}+25s + 42 \end{align*}$$

\]

Step 2: Add the two expanded expressions together

\[

$$\begin{align*} &(6s^{2}+16s + 10)+(3s^{2}+25s + 42)\\ &=6s^{2}+3s^{2}+16s+25s + 10 + 42\\ &=9s^{2}+41s+52 \end{align*}$$

\]

Part (b)

We need to expand and simplify \((2x + 3)(5x + 4)+(x - 4)(3x - 7)\)

Step 1: Expand each product using the distributive property (FOIL method)

• Expand \((2x + 3)(5x + 4)\):

\[

$$\begin{align*} (2x + 3)(5x + 4)&=2x\times5x+2x\times4+3\times5x + 3\times4\\ &=10x^{2}+8x+15x + 12\\ &=10x^{2}+23x + 12 \end{align*}$$

\]

• Expand \((x - 4)(3x - 7)\):

\[

$$\begin{align*} (x - 4)(3x - 7)&=x\times3x-x\times7-4\times3x+4\times7\\ &=3x^{2}-7x-12x + 28\\ &=3x^{2}-19x + 28 \end{align*}$$

\]

Step 2: Add the two expanded expressions together

\[

$$\begin{align*} &(10x^{2}+23x + 12)+(3x^{2}-19x + 28)\\ &=10x^{2}+3x^{2}+23x-19x+12 + 28\\ &=13x^{2}+4x + 40 \end{align*}$$

\]

Final Answers

a) \(\boldsymbol{9s^{2}+41s + 52}\)

b) \(\boldsymbol{13x^{2}+4x + 40}\)