Problem 131 - College Algebra Essentials

Question

Simplify: \(2(x+h)^{2}+3(x+h)+5-\left(2 x^{2}+3 x+5\right)\)

Step-by-Step Solution

Answer

The simplified form of the expression is \(4hx+2h^2+3h-5\)

Step 1: Expansion of the expression

Expand \(2(x+h)^{2}+3(x+h)\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\). This gives us \(2(x^2+2xh+h^2)+3x+3h\), which can be further simplified to \(2x^2+4hx+2h^2+3x+3h\).

Step 2: Subtract the second expression

Subtract \(2 x^{2}+3 x+5\) from the expanded expression. We have: \(2x^2+4hx+2h^2+3x+3h - (2x^2+3x+5)\). Distribute the negative sign into the parenthesis. The expression becomes \(2x^2+4hx+2h^2+3x+3h - 2x^2 - 3x - 5\).

Step 3: Grouping like terms and simplify

Now, let's group the common terms and simplify the expression. The terms \(2x^2\), \(4hx\), \(2h^2\), \(3x\), \(3h\) are matched with \(-2x^2\), \(-3x\), \(-5\) respectively. This leaves us with the expression \(4hx+2h^2+3h-5\).