Problem 103 - Algebra and Trigonometry

Question

Factor completely. $$10 x^{2}(x+1)-7 x(x+1)-6(x+1)$$

Step-by-Step Solution

Answer

The completely factored form of \(10 x^{2}(x+1)-7 x(x+1)-6(x+1)\) is \((x+1)(5x - 6)(2x + 1)\).

Step 1: Identify the Common Factor

First, determine the common factor for each term of the expression. For this equation, the common factor for all three terms is \((x+1)\). The expression can be rewritten as: \(10x^2*(x+1) - 7x*(x+1) - 6*(x+1)\).

Step 2: Factor Out the Common Factor

The common factor \((x+1)\) can be factored out from each term of the expression: \((x+1)*(10x^2 - 7x - 6)\).

Step 3: Factorize the Quadratic Equation

Now factorize the quadratic part that is still intact. For the quadratic \(10x^2 - 7x - 6\), look for two numbers that multiply to \(-60\) (which is the result of the product of \(10\) and \(-6\)) and add up to \(-7\). Those numbers are \(-12\) and \(5\). The quadratic factorizes to \((5x - 6)(2x + 1)\).

Step 4: Write the Final Factored Form

Now, incorporate this back into the expression to have the fully factored form: \((x+1)(5x - 6)(2x + 1)\).