Factor and simplify each algebraic expression. $-8(4x + 3)^{-2} + 10(5x + 1)(4x + 3)^{-1}$

Verified step by step guidance

1. Identify the common factors in the expression: the terms involve powers of \( (4x+3) \), specifically \( (4x+3)^{-2} \) and \( (4x+3)^{-1} \).
2. Rewrite the expression to clearly see the common factor: \( -8(4x+3)^{-2} + 10(5x+1)(4x+3)^{-1} \). Notice that \( (4x+3)^{-2} = \frac{1}{(4x+3)^2} \) and \( (4x+3)^{-1} = \frac{1}{4x+3} \).
3. Factor out the smallest power of \( (4x+3) \), which is \( (4x+3)^{-2} \), from both terms: \( (4x+3)^{-2} \) times the remaining expression inside parentheses.
4. Inside the parentheses, divide each original term by \( (4x+3)^{-2} \): the first term becomes \( -8 \), and the second term becomes \( 10(5x+1)(4x+3)^{-1 + 2} = 10(5x+1)(4x+3)^1 = 10(5x+1)(4x+3) \).
5. Write the factored expression as \( (4x+3)^{-2} \left[ -8 + 10(5x+1)(4x+3) \right] \). Then, simplify inside the brackets by expanding and combining like terms if possible.

Key Concepts

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent. For example, a^(-n) = 1/a^n. Understanding this helps in rewriting expressions like (4x+3)^-2 as 1/(4x+3)^2, which is essential for simplifying the given expression.

Factoring Algebraic Expressions

Factoring involves rewriting an expression as a product of its factors. Recognizing common factors, such as (4x+3)^-1 in this problem, allows you to simplify complex expressions by factoring out shared terms, making the expression easier to combine and reduce.

Combining Like Terms with Rational Expressions

When expressions involve terms with variables raised to powers or negative exponents, combining like terms requires expressing them with a common denominator or base. This skill is crucial for adding or subtracting terms like those in the problem, enabling simplification into a single, reduced expression.