1. **State the problem:** Factor each quadratic expression given. 2. **Recall the factoring formulas and rules:** - For quadratics of the form $ax^2 + bx + c$, look for two numbers that multiply to $ac$ and add to $b$. - For difference of squares: $a^2 - b^2 = (a - b)(a + b)$. - For perfect square trinomials: $a^2 \\pm 2ab + b^2 = (a \\pm b)^2$. 3. **Factor each expression step-by-step:** **a.** $4x^2 + 15$ cannot be factored further over the integers because it is a sum, not difference, and 15 is not a perfect square. **b.** $y^2 + 12y + 35$ - Find two numbers that multiply to 35 and add to 12: 7 and 5. - Factor as $(y + 7)(y + 5)$. **c.** $z^2 + z - 6$ - Find two numbers that multiply to -6 and add to 1: 3 and -2. - Factor as $(z + 3)(z - 2)$. **d.** $x^2 - 6x + 8$ - Find two numbers that multiply to 8 and add to -6: -4 and -2. - Factor as $(x - 4)(x - 2)$. **e.** $x^2 - 16$ - Recognize difference of squares: $x^2 - 4^2$. - Factor as $(x - 4)(x + 4)$. **f.** $3x^2 + 7x + 2$ - Multiply $a imes c = 3 imes 2 = 6$. - Find two numbers that multiply to 6 and add to 7: 6 and 1. - Rewrite middle term: $3x^2 + 6x + x + 2$. - Factor by grouping: $$3x(x + 2) + 1(x + 2) = (3x + 1)(x + 2)$$ **g.** $2x^2 + x - 21$ - Multiply $a imes c = 2 imes (-21) = -42$. - Find two numbers that multiply to -42 and add to 1: 7 and -6. - Rewrite middle term: $2x^2 + 7x - 6x - 21$. - Factor by grouping: $$x(2x + 7) - 3(2x + 7) = (x - 3)(2x + 7)$$ **h.** $10x^2 + 9x + 2$ - Multiply $a imes c = 10 imes 2 = 20$. - Find two numbers that multiply to 20 and add to 9: 5 and 4. - Rewrite middle term: $10x^2 + 5x + 4x + 2$. - Factor by grouping: $$5x(2x + 1) + 2(2x + 1) = (5x + 2)(2x + 1)$$ **i.** $12x^2 + 17x - 14$ - Multiply $a imes c = 12 imes (-14) = -168$. - Find two numbers that multiply to -168 and add to 17: 21 and -8. - Rewrite middle term: $12x^2 + 21x - 8x - 14$. - Factor by grouping: $$3x(4x + 7) - 2(4x + 7) = (3x - 2)(4x + 7)$$ **j.** $4a^2 - 9$ - Recognize difference of squares: $(2a)^2 - 3^2$. - Factor as $(2a - 3)(2a + 3)$. 4. **Final factored forms:** - $4x^2 + 15$ (prime over integers) - $(y + 7)(y + 5)$ - $(z + 3)(z - 2)$ - $(x - 4)(x - 2)$ - $(x - 4)(x + 4)$ - $(3x + 1)(x + 2)$ - $(x - 3)(2x + 7)$ - $(5x + 2)(2x + 1)$ - $(3x - 2)(4x + 7)$ - $(2a - 3)(2a + 3)$