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Problem 138

Add the expressions $$4 a^{2}-3+5 a, 6 a-2 a^{2}+2$$, and $$2 a^{2}-3 a+8$$

Expert verified
The sum of the expressions $$4a^2 - 3 + 5a$$, $$6a - 2a^2 + 2$$, and $$2a^2 - 3a + 8$$ is $$4a^2 + 8a + 7$$.
See the step by step solution

Step 1: Identify the like terms

In the given expressions, the like terms are those with the same variable raised to the same power. For our expressions, the like terms are: - $$4a^2, -2a^2$$, and $$2a^2$$ (all are terms with $$a^2$$) - $$5a$$ and $$6a$$, and $$-3a$$ (all are terms with $$a$$) - $$-3, 2$$, and $$8$$ (all are constant terms)

Step 2: Add the coefficients of like terms

Now we will add the coefficients of the like terms together: - For the terms with $$a^2$$: $$4a^2 - 2a^2 + 2a^2 = (4 - 2 + 2)a^2 = 4a^2$$ - For the terms with $$a$$: $$5a + 6a - 3a = (5 + 6 - 3)a = 8a$$ - For the constant terms: $$-3 + 2 + 8 = -3 + 10 = 7$$

Step 3: Write the sum of the expressions

Combine the resulting terms to get the sum of the given expressions: $$4a^2 + 8a + 7$$ So, the sum of the expressions $$4a^2 - 3 + 5a$$, $$6a - 2a^2 + 2$$, and $$2a^2 - 3a + 8$$ is $$4a^2 + 8a + 7$$.

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