Factor completely: 4m² + m - 5

Understand the Problem

The question asks us to factor the quadratic expression completely. The expression given is 4m² + m - 5, and we need to find its factors.

Answer

The expression factors to $(4m + 5)(m - 1)$.

Steps to Solve

Identify the coefficients The given expression is $4m^2 + m - 5$. We identify the coefficients:
- (a = 4) (coefficient of $m^2$)
- (b = 1) (coefficient of $m$)
- (c = -5) (constant term)
Use the product-sum method We look for two numbers that multiply to (a \cdot c = 4 \cdot (-5) = -20) and add up to (b = 1). The pair that fits these criteria is (5) and (-4), since:
- (5 \times (-4) = -20)
- (5 + (-4) = 1)
Rewrite the middle term We can rewrite the expression using these numbers: $$4m^2 + 5m - 4m - 5$$
Group the terms Now, we group the terms: $$(4m^2 + 5m) + (-4m - 5)$$
Factor by grouping Factor out the common factors in each group: $$m(4m + 5) - 1(4m + 5)$$
Factor out the common binomial Now, we can factor out the common binomial factor: $$(4m + 5)(m - 1)$$

The completely factored form of the expression is $(4m + 5)(m - 1)$.

More Information

Factoring quadratics is a fundamental algebra skill. It involves breaking down an expression into simpler parts that, when multiplied, give the original expression. This technique is useful for solving equations, simplifying expressions, and graphing quadratic functions.

Tips

Not checking the signs: Ensure the signs are correct when identifying factor pairs.
Forgetting to factor out common terms: Always look for common factors in grouped expressions.
Assuming all quadratics can be factored easily: Some quadratics may require the quadratic formula if they don't factor nicely.

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