-x^2- Chapter 2.5 Review Factor completely. If the polynomial is not factorable, write prime. 31. x^2+10x-39 32. x^2-16 33. 8x^3+27 34. 18x^3+3x^2-3x (x+8)(x+8) 36. 9x^2-30x+16 37. 25x^2+36 39. 4x^2+13x-12

## Step 1: Factoring $x^2 + 10x - 39$<br />### We look for two numbers that multiply to -39 and add up to 10. These numbers are 13 and -3. Thus, the factored form is $(x+13)(x-3)$.<br /><br />## Step 2: Factoring $x^2 - 16$<br />### This is a difference of squares, $x^2 - 4^2$. The factored form is $(x+4)(x-4)$.<br /><br />## Step 3: Factoring $8x^3 + 27$<br />### This is a sum of cubes, $(2x)^3 + 3^3$. The factored form is $(2x+3)(4x^2 - 6x + 9)$.<br /><br />## Step 4: Factoring $18x^3 + 3x^2 - 3x$<br />### First, factor out the greatest common factor, $3x$: $3x(6x^2 + x - 1)$. Then, factor the quadratic: $3x(3x-1)(2x+1)$.<br /><br />## Step 5: Factoring $9x^2 - 30x + 16$<br />### We are looking for two numbers that multiply to $9 \cdot 16 = 144$ and add up to -30. These numbers are -24 and -6. Rewrite the middle term and factor by grouping: $9x^2 - 24x - 6x + 16 = 3x(3x-8) - 2(3x-8) = (3x-2)(3x-8)$. Alternatively, recognize this as a perfect square trinomial close to $(3x-4)^2$. Expanding that gives $9x^2 - 24x + 16$. Since we have $-30x$ instead of $-24x$, it is not a perfect square trinomial.<br /><br />## Step 6: Factoring $25x^2 + 36$<br />### This is a sum of squares and cannot be factored further using real numbers. Thus, it is prime.<br /><br />## Step 7: Factoring $4x^2 + 13x - 12$<br />### We look for two numbers that multiply to $4 \cdot -12 = -48$ and add up to 13. These numbers are 16 and -3. Rewrite the middle term and factor by grouping: $4x^2 + 16x - 3x - 12 = 4x(x+4) - 3(x+4) = (4x-3)(x+4)$.