Question
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Tutor-Verified Answer
Answer
The simplified form of the expression is \(3x^{3}-10x^{2}-27x+10\).
Solution
Calculate or simplify the expression \( 3x^2(x+2)-16x(x+2)+5(x+2) \).
Simplify the expression by following steps:
- step0: Solution:
\(3x^{2}\left(x+2\right)-16x\left(x+2\right)+5\left(x+2\right)\)
- step1: Expand the expression:
\(3x^{3}+6x^{2}-16x\left(x+2\right)+5\left(x+2\right)\)
- step2: Expand the expression:
\(3x^{3}+6x^{2}-16x^{2}-32x+5\left(x+2\right)\)
- step3: Expand the expression:
\(3x^{3}+6x^{2}-16x^{2}-32x+5x+10\)
- step4: Subtract the terms:
\(3x^{3}-10x^{2}-27x+10\)
The simplified form of the expression \(3x^{2}(x+2)-16x(x+2)+5(x+2)\) is \(3x^{3}-10x^{2}-27x+10\).
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Simplify this solution The Deep Dive
To solve the expression \( 3x^{2}(x+2)-16x(x+2)+5(x+2) \), let's first factor out the common term \((x+2)\): \[ (x+2)(3x^{2}-16x+5) \] Now we will focus on factoring the quadratic \(3x^{2}-16x+5\). To factor this, we look for two numbers that multiply to \(3 \cdot 5 = 15\) and add up to \(-16\). These numbers are \(-15\) and \(-1\): \[ 3x^{2}-15x-x+5 = 3x(x-5)-1(x-5) \] Thus, it factors to: \[ (3x-1)(x-5) \] Putting it all together, we can write the original expression as: \[ (x+2)(3x-1)(x-5) \] This simplifies the problem and gives us the complete factored form of the expression.
