376. \( [12 x y-(-7 x y+13 x y)-5 x y]^{2}:\left(-3 x^{2}\right)-(-2 y)^{2} \)

To simplify the expression \( 376. \, [12xy - (-7xy + 13xy) - 5xy]^2 : (-3x^2) - (-2y)^2 \), let's break it down step by step. First, simplify the expression inside the brackets: 1. Combine the terms: \[ -7xy + 13xy = 6xy \] So, we have: \[ 12xy - 6xy - 5xy \] 2. Now, facilitating further simplification: \[ 12xy - 6xy - 5xy = (12xy - 6xy) - 5xy = 6xy - 5xy = xy \] 3. Now, substitute back: \[ [xy]^2 = x^2y^2 \] Now look at the full expression: 4. We substitute back into our original format: \[ 376 \cdot x^2y^2 : (-3x^2) - (-2y)^2 \] 5. Calculate \((-2y)^2\): \[ (-2y)^2 = 4y^2 \] 6. The expression now reads: \[ \frac{376x^2y^2}{-3x^2} - 4y^2 \] Now simplifying the fraction: 7. Simplify: \[ 376x^2y^2 \div -3x^2 = -\frac{376y^2}{3} \] So, putting it all together we have: \[ -\frac{376y^2}{3} - 4y^2 \] Next, we need a common denominator to combine the terms: - The common denominator between \( -\frac{376y^2}{3} \) and \( -4y^2 \) (which can be expressed as \( -\frac{12y^2}{3} \)). Combining: \[ -\frac{376y^2}{3} - \frac{12y^2}{3} = -\frac{376y^2 + 12y^2}{3} = -\frac{388y^2}{3} \] Final result: \[ -\frac{388y^2}{3} \]