### Worked example 1: Multiplying out two sets of brackets (Rectangle area method)

- Notes
- Expand and simplify $ \big(x+2\big)\big(x+1\big)$
- \[\begin{array}{l} (x + 2)(x + 1)\\ = {x^2} + 2x + 1x + 2\\ = {x^2} + 3x + 2 \end{array}\]
The first bracket is represented by the horizontal length in the diagram, and the 2nd bracket is represented by the vertical width of the rectangle.

Multiplying the length by width to find the area of the rectangle gives us the area of the rectangle.

It is a good idea to put the signs in. In this example there are no negatives, but in the next example, there are and it can get quite confusing, if you are not careful.

- $x^2+3x+1$

### Worked example 2: Multiplying out complex brackets (Rectangle area method)

- Notes
Expand and simplify the expression $(3c^2d - 4d+1)(2cd^2+2d-1)$

- \[\begin{array}{l} (2{c^2}d - 4d + 1)({c^2}d + 2d - 1)\\ = 2{c^4}{d^2} - 4{c^2}{d^3} + 3{c^2}d + 4{c^2}{d^2} - 8{d^2} + 6d - 2{c^2}d + 4d - 3\\ = 2{c^4}{d^2} - 4{c^2}{d^3} + 4{c^2}{d^2} + {c^2}d - 8{d^2} + 10d - 3 \end{array}\]
Problems of this sort can be tackled in a variety of ways, but it is always important to be systematic. If you try to deal with things like this in a loose manner, the likelihood of making errors, increase substantially.

One approach is to use a rectangular diagram like the one to the left. Each bracket is represented by one dimension of the rectangle. Note that it is helpful to place the correct sign before each term.

Technically, you might think that finding areas likes this is a bit silly if the lengths are negative. However, it does work well nonetheless. Even though it makes no practical sense, mathematically, it is fine and dandy.

There are only two pairs of terms which can be combined, the $+4d$ and the $+6d$ and the terms in $c^2d$. Once they are combined, we are finished.

- \[2{c^4}{d^2} - 4{c^2}{d^3} + 4{c^2}{d^2} + {c^2}d - 8{d^2} + 10d - 3\]

### Exercise 1: Multiplying 2 brackets together (All signs +ve)

**Instructions:**Expand and simplify the following brackets:

- Q1\[(s+6)(s+4)\]$s^2+10s+24$
- Q2\[(h+3)(h+5)\]$h^2+8h+15$
- Q3\[(b+3)(b+4)\]$b^2+7b+12$
- Q4\[(m+2)(m+3)\]$m^2+5m+6$
- Q5\[(n+4)(n+5)\]$n^2+9b+20$
- Q6\[(c+7)(c+3)\]$c^2+10c+21$
- Q7\[(i+7)(i+9)\]$i^2+16i+63$
- Q8\[(f+1)(f+2)\]$f^2+3f+2$
- Q9\[(d+5)(d+8)\]$d^2+13s+40$
- Q10\[(e+8)(e+9)\]$e^2+17e+72$

### Exercise 2: Multiplying 2 brackets together (All signs -ve)

**Instructions:**Expand and simplify the following expressions:

Remember : a negative number multiplied by a negative number is a positive number.

- Q1\[(j-2)(j-4)\]$j^2-6j+8$
- Q2\[(i-1)(i-3)\]$i^2-4i+3$
- Q3\[(u-7)(u-3)\]$u^2-10y+21$
- Q4\[(d-2)(d-5)\]$d^2-7d+10$
- Q5\[(k-6)(k-4)\]$k^2-10k+24$
- Q6\[(w-4)(w-5)\]$w^2-9w+20$
- Q7\[(z-2)(z-3)\]$z^2-5z+6$
- Q8\[(e-8)(e-3)\]$e^2-11e+24$
- Q9\[(x-1)(x-2)\]$x^2-3x+2$
- Q10\[(g-7)(g-9)\]$g^2-16g+63$

### Exercise 3: Squaring brackets

**Instructions:**Expand and simplify the following expressions:

- Q1\[(r+6)^2\]$r^2+12r+36$
- Q2\[(w-9)^2\]$w^2-18w+81$
- Q3\[(v+5)^2\]$v^2+10v+25$
- Q4\[(q-2)^2\]$q^2-4q+4$
- Q5\[(k-3)^2\]$k^2-6k+9$
- Q6\[(h-6)^2\]$h^2-12h+36$
- Q7\[(z+9)^2\]$z^2+18z+81$
- Q8\[(m-7)^2\]$m^2-14m+49$
- Q9\[(g-5)^2\]$g^2-10g+25$
- Q10\[(a+2)^2\]$a^2+4a+4$

### Exercise 4: Multiplying 2 brackets together (Signs mixed) Pattern spotting

**Instructions:**Expand and simplify the following expressions:

Try to spot the pattern/similarity between each pair of questions.

- Q1\[(u+2)(u-4)\]$u^2-2u-8$
- Q2\[(k-6)(k+2)\]$k^2-4k-12$
- Q3\[(q+1)(q-3)\]$q^2-2q-3$
- Q4\[(r+5)(r-3)\]$r^2+2r-15$
- Q5\[(b+7)(b-5)\]$b^2+2b-35$
- Q6\[(q-1)(q+3)\]$q^2+2q-3$
- Q7\[(e+1)(e-2)\]$e^2-e-2$
- Q8\[(e-1)(e+2)\]$e^2+e-2$
- Q9\[(u-2)(u+4)\]$u^2+2u-8$
- Q10\[(k+6)(k-2)\]$k^2+4k-12$

### Exercise 5: Multiplying two brackets (Signs mixed)

**Instructions:**Expand and simplify the following expressions:

- Q1

### Exercise 6: Difference of two squares $ \big(x-a \big) \big(x+a \big)$

**Instructions:**Expand and simplify the following expressions:

Try to spot the pattern in these examples. What do they all have in common?

- Q1

### Exercise 7: Multiplying brackets (Mixed exercise)

**Instructions:**Expand and simplify the following expressions.

Take your time over these as they are all different. Try to identify which

**type**of questions they are.- Q1\[(b+6 )(b+5 )\]$b^2+11b+30$
- Q2\[(y-3 )(y-9 )\]$y^2-12y+27$
- Q3\[(a+3 )(a+7 )\]$a^2+10a+21$