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When dealing with polynomial subtraction, one of the key steps is distributing the negative sign across the terms inside any parentheses. This step ensures that each term is appropriately influenced by the subtraction operation. Let's see how it works in practice with the expression \[(8x+4)-(5x-1)-(7x-5).\]The negative sign in front of a parenthesis basically acts as a multiplier of \(-1\) for all terms inside it. For example, when we distribute the negative sign across \((5x-1)\), it becomes \(-5x + 1\). Similarly, distributing a negative sign across \((7x-5)\) results in \(-7x + 5\). Therefore, the expression transforms into:

• \(8x + 4 - 5x + 1\)
• \(- 7x + 5\)

This rearrangement allows us to easily combine the terms in the next step, maintaining the integrity of the original mathematical operation.

Once the negative signs have been distributed across the polynomial, the next step is to **combine like terms**. This step simplifies the expression by merging terms that share the same variable or are constant values.In the expression \(8x + 4 - 5x + 1 - 7x + 5\), like terms must be identified and added together. Here, it can be broken down into two categories:

• Terms with \(x\): These include \(8x\), \(-5x\), and \(-7x\).
• Constant terms: These include \(4\), \(1\), and \(5\).

Now, we perform the addition/subtraction:

• Combine the \(x\) terms: \(8x - 5x - 7x = -4x\).
• Combine the constants: \(4 + 1 + 5 = 10\).

By combining like terms, we simplify the polynomial expression, getting us closer to the final answer.

The fruit of your labor comes in the final step, where you write the simplified version of the polynomial, known as the **final expression**. After distributing negative signs and combining like terms, all that's left is to present the final clean version of the equation.For our example, the expression combined down to:

• Terms involving \(x\): \(-4x\)
• Constant terms: \(+ 10\)

Put them together, and your final polynomial expression becomes:\[-4x + 10\]This result is a simplified form of the original polynomial operation. Assessing your work, ensure there are no further like terms to combine. This equation can now be used for further algebraic operations or left as the answer to the problem.