Question 1

We will factorize each given quadratic expression into its binomial factors.

Step 1 — Factorize $s^2 - 11s + 24$

We need two numbers. Their product must be 24. Their sum must be -11. These numbers are -3 and -8.

$s^2 - 11s + 24$

$= s^2 - 3s - 8s + 24$

$= s(s - 3) - 8(s - 3)$

$\boxed{(s - 3)(s - 8)}$

Step 2 — Find the missing factor for $(x + 1)$

We need to factor the expression $3x^2 - 4x - 7$. We look for two numbers. Their product must be $3 \times -7 = \textbf{-21}$. Their sum must be -4. These numbers are -7 and 3.

$3x^2 - 4x - 7$

$= 3x^2 - 7x + 3x - 7$

$= x(3x - 7) + 1(3x - 7)$

$\boxed{(3x - 7)(x + 1)}$

Step 3 — Find the missing terms for $10x^2 - 11x - 6$

We need to factor the expression $10x^2 - 11x - 6$. We look for two numbers. Their product must be $10 \times -6 = \textbf{-60}$. Their sum must be -11. These numbers are -15 and 4.

$10x^2 - 11x - 6$

$= 10x^2 - 15x + 4x - 6$

$= 5x(2x - 3) + 2(2x - 3)$

$\boxed{(2x - 3)(5x + 2)}$

Step 4 — Factorize $6x^2 + 7x + 2$

We need two numbers. Their product must be $6 \times 2 = \textbf{12}$. Their sum must be 7. These numbers are 3 and 4.

$6x^2 + 7x + 2$

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

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

$\boxed{(3x + 2)(2x + 1)}$

Answer

(i) $s^2 - 11s + 24 = (s - 3)(s - 8)$ (ii) $(3x - 7)(x + 1) = (3x^2 - 4x - 7)$ (iii) $10x^2 - 11x - 6 = (2x - 3)(5x + 2)$ (iv) $6x^2 + 7x + 2 = (3x + 2)(2x + 1)$