1. **Problem Statement:** (a) Factorize completely: (i) $a^2 - 25$ (ii) $2m^2 + 3m - 5$ (b) Given the operation $a * b = 4a - b$, find: (i) $3 * 2$ (ii) $p$ if $p * 1 = 19$ 2. **Formula and Rules:** - Difference of squares: $x^2 - y^2 = (x - y)(x + y)$ - Quadratic factorization: Find two numbers that multiply to $ac$ and add to $b$ in $ax^2 + bx + c$ - Use the given custom operation definition for part (b) 3. **Step-by-step Solution:** (a)(i) Factorize $a^2 - 25$: - Recognize this as a difference of squares: $a^2 - 5^2$ - Apply formula: $a^2 - 25 = (a - 5)(a + 5)$ (a)(ii) Factorize $2m^2 + 3m - 5$: - Multiply $a$ and $c$: $2 imes (-5) = -10$ - Find two numbers that multiply to $-10$ and add to $3$: $5$ and $-2$ - Rewrite middle term: $2m^2 + 5m - 2m - 5$ - Group terms: $(2m^2 + 5m) + (-2m - 5)$ - Factor each group: $m(2m + 5) -1(2m + 5)$ - Factor out common binomial: $(m - 1)(2m + 5)$ (b)(i) Calculate $3 * 2$: - Use definition: $3 * 2 = 4(3) - 2 = 12 - 2 = 10$ (b)(ii) Find $p$ such that $p * 1 = 19$: - Use definition: $p * 1 = 4p - 1 = 19$ - Solve for $p$: $4p - 1 = 19$ - Add 1 to both sides: $4p = 20$ - Divide both sides by 4: $p = 5$ 4. **Final Answers:** (a)(i) $(a - 5)(a + 5)$ (a)(ii) $(m - 1)(2m + 5)$ (b)(i) $10$ (b)(ii) $5$