To expand the expression \((4x + 5)(x^2 - 2x + 5)\), we will use the distributive property (also known as the FOIL method for binomials). Each term in the first polynomial will be multiplied by each term in the second polynomial.
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Distributing \(4x\): \[ 4x \cdot x^2 = 4x^3 \] \[ 4x \cdot (-2x) = -8x^2 \] \[ 4x \cdot 5 = 20x \]
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Distributing \(5\): \[ 5 \cdot x^2 = 5x^2 \] \[ 5 \cdot (-2x) = -10x \] \[ 5 \cdot 5 = 25 \]
Now, we can combine all these results: \[ 4x^3 - 8x^2 + 20x + 5x^2 - 10x + 25 \]
Next, we combine the like terms:
- The \(x^2\) terms: \(-8x^2 + 5x^2 = -3x^2\)
- The \(x\) terms: \(20x - 10x = 10x\)
Putting it all together, the expanded polynomial in standard form is: \[ 4x^3 - 3x^2 + 10x + 25 \]
Thus, the final answer is: \[ \boxed{4x^3 - 3x^2 + 10x + 25} \]
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