Step 1: Factorize the denominator. The denominator is $9x^2 - y^2$. This is a difference of squares, which follows the formula $a^2 - b^2 = (a-b)(a+b)$. Here, $a = 3x$ and $b = y$. $$9x^2 - y^2 = (3x)^2 - y^2 = (3x - y)(3x + y)$$ Step 2: Attempt to factorize the numerator. The numerator is $3x^2 - 4xy - y^2$. We look for two binomials $(Ax+By)(Cx+Dy)$ such that their product is $3x^2 - 4xy - y^2$. Expanding $(Ax+By)(Cx+Dy) = ACx^2 + (AD+BC)xy + BDy^2$. We need to find integer coefficients $A, B, C, D$ such that: $AC = 3$ $BD = -1$ $AD+BC = -4$ Possible integer pairs for $(A,C)$: $(1,3)$ or $(3,1)$. Possible integer pairs for $(B,D)$: $(1,-1)$ or $(-1,1)$. Let's test all combinations: • If $(A,C) = (1,3)$ and $(B,D) = (1,-1)$: $(x+y)(3x-y) = 3x^2 - xy + 3xy - y^2 = 3x^2 + 2xy - y^2$. (Incorrect middle term) • If $(A,C) = (1,3)$ and $(B,D) = (-1,1)$: $(x-y)(3x+y) = 3x^2 + xy - 3xy - y^2 = 3x^2 - 2xy - y^2$. (Incorrect middle term) • If $(A,C) = (3,1)$ and $(B,D) = (1,-1)$: $(3x+y)(x-y) = 3x^2 - 3xy + xy - y^2 = 3x^2 - 2xy - y^2$. (Incorrect middle term) • If $(A,C) = (3,1)$ and $(B,D) = (-1,1)$: $(3x-y)(x+y) = 3x^2 + 3xy - xy - y^2 = 3x^2 + 2xy - y^2$. (Incorrect middle term) Since none of these combinations result in the middle term $-4xy$, the numerator $3x^2 - 4xy - y^2$ is not factorable over integers. Step 3: Write the final factored expression. Since the numerator cannot be factored over integers, the expression is presented with the denominator factored. The fully factorized expression is: $$\frac{3x^2 - 4xy - y^2}{(3x - y)(3x + y)}$$ The numerator $3x^2 - 4xy - y^2$ is not factorable over integers. The final answer is $\boxed{\frac{3x^2 - 4xy - y^2}{(3x - y)(3x + y)}}$. Send me the next one 📸