Question

1. $4b(5b - 3) - 2(b^2 - 7b - 4)$
2. $4n(3n^2 + n - 4) - n(3 - n)$
3. $3m(3m + 6) - 3(m^2 + 4m + 1)$
4. $2(4k^2 - 2k) - 3(-6k^2 + 4) + 2k(k - 1)$
5. write an expression in simplest form to represent the area of the shaded region.

$3x + 2$
$4x$
$3x$
$2x$

Explanation:

Step1: Simplify Question 16

Expand and combine like terms.

$$\begin{align*} 4b(5b-3)-2(b^2-7b-4)&=20b^2-12b-2b^2+14b+8\\ &=(20b^2-2b^2)+(-12b+14b)+8\\ &=18b^2+2b+8 \end{align*}$$

Step2: Simplify Question 17

Expand and combine like terms.

$$\begin{align*} 4n(3n^2+n-4)-n(3-n)&=12n^3+4n^2-16n-3n+n^2\\ &=12n^3+(4n^2+n^2)+(-16n-3n)\\ &=12n^3+5n^2-19n \end{align*}$$

Step3: Simplify Question 18

Expand and combine like terms.

$$\begin{align*} 3m(3m+6)-3(m^2+4m+1)&=9m^2+18m-3m^2-12m-3\\ &=(9m^2-3m^2)+(18m-12m)-3\\ &=6m^2+6m-3 \end{align*}$$

Step4: Simplify Question 19

Expand and combine like terms.

$$\begin{align*} 2(4k^2-2k)-3(-6k^2+4)+2k(k-1)&=8k^2-4k+18k^2-12+2k^2-2k\\ &=(8k^2+18k^2+2k^2)+(-4k-2k)-12\\ &=28k^2-6k-12 \end{align*}$$

Step5: Solve Question 20

Find area of outer rectangle minus inner rectangle.

$$\begin{align*} \text{Area of outer rectangle}&=4x(3x+2)=12x^2+8x\\ \text{Area of inner rectangle}&=3x(2x)=6x^2\\ \text{Shaded Area}&=(12x^2+8x)-6x^2=6x^2+8x \end{align*}$$

Answer:

1. $\boldsymbol{18b^2+2b+8}$
2. $\boldsymbol{12n^3+5n^2-19n}$
3. $\boldsymbol{6m^2+6m-3}$
4. $\boldsymbol{28k^2-6k-12}$
5. $\boldsymbol{6x^2+8x}$