Expand the expression to a polynomial in standard form: \[ \left(2 x^{2}+x-2\right)\left(x^{2}-3 x-6\right) \]

To expand the expression \((2x^{2} + x - 2)(x^{2} - 3x - 6)\), we will use the distributive property (also known as the FOIL method for binomials). 1. Multiply \(2x^{2}\) by each term in \((x^{2} - 3x - 6)\): - \(2x^{2} \cdot x^{2} = 2x^{4}\) - \(2x^{2} \cdot (-3x) = -6x^{3}\) - \(2x^{2} \cdot (-6) = -12x^{2}\) 2. Multiply \(x\) by each term in \((x^{2} - 3x - 6)\): - \(x \cdot x^{2} = x^{3}\) - \(x \cdot (-3x) = -3x^{2}\) - \(x \cdot (-6) = -6x\) 3. Multiply \(-2\) by each term in \((x^{2} - 3x - 6)\): - \(-2 \cdot x^{2} = -2x^{2}\) - \(-2 \cdot (-3x) = 6x\) - \(-2 \cdot (-6) = 12\) Now, let's combine all these results: \[ 2x^{4} + (-6x^{3} + x^{3}) + (-12x^{2} - 3x^{2} - 2x^{2}) + (-6x + 6x) + 12 \] This simplifies to: \[ 2x^{4} - 5x^{3} - 17x^{2} + 12 \] Thus, the expanded expression in standard form is: \[ \boxed{2x^{4} - 5x^{3} - 17x^{2} + 12} \]