Expand the expression to a polynomial in standard form: \[ \left(3 x^{2}+x-4\right)\left(x^{2}-x+4\right) \]

To expand the expression \(\left(3 x^{2}+x-4\right)\left(x^{2}-x+4\right)\), we will use the distributive property (also known as the FOIL method for binomials). We will multiply each term in the first polynomial by each term in the second polynomial. 1. Multiply \(3x^2\) by each term in \(x^2 - x + 4\): - \(3x^2 \cdot x^2 = 3x^4\) - \(3x^2 \cdot (-x) = -3x^3\) - \(3x^2 \cdot 4 = 12x^2\) 2. Multiply \(x\) by each term in \(x^2 - x + 4\): - \(x \cdot x^2 = x^3\) - \(x \cdot (-x) = -x^2\) - \(x \cdot 4 = 4x\) 3. Multiply \(-4\) by each term in \(x^2 - x + 4\): - \(-4 \cdot x^2 = -4x^2\) - \(-4 \cdot (-x) = 4x\) - \(-4 \cdot 4 = -16\) Now, let's combine all the results from each part: \[ 3x^4 + (-3x^3 + x^3) + (12x^2 - x^2 - 4x^2) + (4x + 4x) - 16 \] Combine like terms: - \(x^4\) terms: \(3x^4\) - \(x^3\) terms: \(-3x^3 + x^3 = -2x^3\) - \(x^2\) terms: \(12x^2 - x^2 - 4x^2 = 7x^2\) - \(x\) terms: \(4x + 4x = 8x\) - Constant term: \(-16\) Putting it all together, we get: \[ 3x^4 - 2x^3 + 7x^2 + 8x - 16 \] Thus, the expression in standard form is: \[ 3x^4 - 2x^3 + 7x^2 + 8x - 16 \]