1. Problem: Factorize the expression $x^2 + 5x + 6$. 2. To factorize, find two numbers that multiply to $6$ and add to $5$. 3. The numbers $2$ and $3$ satisfy these conditions: $2 \times 3 = 6$ and $2 + 3 = 5$. 4. Hence, the factorized form is $$(x + 2)(x + 3)$$. 1. Problem: Factorize the difference of squares $a^2 - 16$. 2. Recognize that $16 = 4^2$, so we have $a^2 - 4^2$. 3. Apply the difference of squares formula: $x^2 - y^2 = (x - y)(x + y)$. 4. Thus, $a^2 - 16 = (a - 4)(a + 4)$. 1. Problem: Factorize the quadratic $3x^2 + 11x + 6$. 2. Multiply the coefficient of $x^2$ (3) by the constant term (6): $3 \times 6 = 18$. 3. Find two numbers that multiply to $18$ and add to $11$: these are $9$ and $2$. 4. Rewrite the middle term using these numbers: $3x^2 + 9x + 2x + 6$. 5. Group terms: $(3x^2 + 9x) + (2x + 6)$. 6. Factor out common terms: $3x(x + 3) + 2(x + 3)$. 7. Factor the common binomial: $(3x + 2)(x + 3)$. 1. Problem: Factorize $x^3 - 8$. 2. Recognize this is a difference of cubes with $x^3 - 2^3$. 3. Use the formula for difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. 4. Substitute $a = x$ and $b = 2$: $$(x - 2)(x^2 + 2x + 4)$$. 1. Problem: Factorize $2x^2 - 7x + 3$. 2. Multiply $2 \times 3 = 6$. 3. Find two numbers that multiply to $6$ and add to $-7$: $-6$ and $-1$. 4. Rewrite the middle term: $2x^2 - 6x - x + 3$. 5. Group: $(2x^2 - 6x) - (x - 3)$. 6. Factor each group: $2x(x - 3) - 1(x - 3)$. 7. Factor out common binomial: $(x - 3)(2x - 1)$. Final answers: 1. $(x + 2)(x + 3)$ 2. $(a - 4)(a + 4)$ 3. $(3x + 2)(x + 3)$ 4. $(x - 2)(x^2 + 2x + 4)$ 5. $(x - 3)(2x - 1)$