Use algebra tiles to find $(-x-6)-\left(-x^2+4\right)$.
Combine like terms. For example, write 3 instead of $1+1+1$.
To simplify the expression \((-x - 6) - (-x^2 + 4)\), we first identify the individual components. The expression consists of a linear part, \(-x - 6\), and a quadratic part, \(-x^2 + 4\), which we are subtracting. Using algebra tiles, we imagine \(-x^2\)-tiles, \(-x\)-tiles, and negative constant unit tiles. Subtracting a polynomial is equivalent to adding its additive inverse, which means we change the sign of every term inside the second polynomial.
When we subtract the second polynomial \((-x^2 + 4)\), we distribute the negative sign across its terms: \(-(-x^2) = x^2\) and \(-(4) = -4\). Now, the original expression can be rewritten as a addition problem: \(-x - 6 + x^2 - 4\). This approach allows us to group like terms more efficiently.
Now we group the terms by their degree:
Combining these results, we get the \(x^2\) term, the \(-x\) term, and the constant \(-10\). Writing the polynomial in descending order of powers, we obtain \(x^2 - x - 10\).
After simplifying, the expression \((-x - 6) - (-x^2 + 4)\) results in the standard polynomial form: \(\boxed{x^2 - x - 10}\).
Algebra tiles are not just for paper-and-pencil math; they represent a fundamental way to visualize how quantities interact. Imagine you are managing a warehouse where you track positive inventory (incoming stock) and negative inventory (outgoing orders or debts). If you have a current status of \(-x - 6\) and you need to remove a negative order of \(-x^2 + 4\), subtracting this negative order effectively increases your positive assets. Learning to combine these terms is the basis for algebraic modeling in economics, physics, and engineering.