The problem is to factorise the following quadratic expressions: 1. $2x^2 + 5x + 3$

1. Problem Description

The problem is to factorise the following quadratic expressions:

1. $2x^2 + 5x + 3$

2. $2x^2 + 7x + 3$

3. $3x^2 + 7x + 2$

4. $2x^2 + 11x + 12$

5. $3x^2 + 8x + 4$

6. $2x^2 + 7x + 5$

7. $3x^2 - 5x - 2$

8. $2x^2 - x - 15$

9. $2x^2 + x - 21$

2. Solution Steps

1. $2x^2 + 5x + 3$

We need to find two numbers that multiply to $2*3 = 6$ and add to $5$. These are $2$ and $3$.

$2x^2 + 2x + 3x + 3 = 2x(x+1) + 3(x+1) = (2x+3)(x+1)$

2. $2x^2 + 7x + 3$

We need to find two numbers that multiply to $2*3 = 6$ and add to $7$. These are $1$ and $6$.

$2x^2 + x + 6x + 3 = x(2x+1) + 3(2x+1) = (x+3)(2x+1)$

3. $3x^2 + 7x + 2$

We need to find two numbers that multiply to $3*2 = 6$ and add to $7$. These are $1$ and $6$.

$3x^2 + x + 6x + 2 = x(3x+1) + 2(3x+1) = (x+2)(3x+1)$

4. $2x^2 + 11x + 12$

We need to find two numbers that multiply to $2*12 = 24$ and add to $11$. These are $3$ and $8$.

$2x^2 + 3x + 8x + 12 = x(2x+3) + 4(2x+3) = (x+4)(2x+3)$

5. $3x^2 + 8x + 4$

We need to find two numbers that multiply to $3*4 = 12$ and add to $8$. These are $2$ and $6$.

$3x^2 + 2x + 6x + 4 = x(3x+2) + 2(3x+2) = (x+2)(3x+2)$

6. $2x^2 + 7x + 5$

We need to find two numbers that multiply to $2*5 = 10$ and add to $7$. These are $2$ and $5$.

$2x^2 + 2x + 5x + 5 = 2x(x+1) + 5(x+1) = (2x+5)(x+1)$

7. $3x^2 - 5x - 2$

We need to find two numbers that multiply to $3*(-2) = -6$ and add to $-5$. These are $1$ and $-6$.

$3x^2 + x - 6x - 2 = x(3x+1) - 2(3x+1) = (x-2)(3x+1)$

8. $2x^2 - x - 15$

We need to find two numbers that multiply to $2*(-15) = -30$ and add to $-1$. These are $5$ and $-6$.

$2x^2 + 5x - 6x - 15 = x(2x+5) - 3(2x+5) = (x-3)(2x+5)$

9. $2x^2 + x - 21$

We need to find two numbers that multiply to $2*(-21) = -42$ and add to $1$. These are $-6$ and $7$.

$2x^2 - 6x + 7x - 21 = 2x(x-3) + 7(x-3) = (2x+7)(x-3)$

3. Final Answer

1. $(2x+3)(x+1)$

2. $(x+3)(2x+1)$

3. $(x+2)(3x+1)$

4. $(x+4)(2x+3)$

5. $(x+2)(3x+2)$

6. $(2x+5)(x+1)$

7. $(x-2)(3x+1)$

8. $(x-3)(2x+5)$

9. $(2x+7)(x-3)$