This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Understand the given function. We are given the function h(x) = x^2 + 2x - 4. Step 2: Solve part a). a) Find h(b+a). Substitute x = (b+a) into the function: h(b+a) = (b+a)^2 + 2(b+a) - 4 Expand the terms: h(b+a) = (b^2 + 2ab + a^2) + (2b + 2a) - 4 h(b+a) = a^2 + b^2 + 2ab + 2a + 2b - 4 The simplified expression is a^2 + b^2 + 2ab + 2a + 2b - 4. Step 3: Solve part b). b) Find h(a-b). Substitute x = (a-b) into the function: h(a-b) = (a-b)^2 + 2(a-b) - 4 Expand the terms: h(a-b) = (a^2 - 2ab + b^2) + (2a - 2b) - 4 h(a-b) = a^2 + b^2 - 2ab + 2a - 2b - 4 The simplified expression is a^2 + b^2 - 2ab + 2a - 2b - 4. Step 4: Solve part c). c) Find (h(1+b)-h(1))/(b). First, find h(1+b): h(1+b) = (1+b)^2 + 2(1+b) - 4 h(1+b) = (1 + 2b + b^2) + (2 + 2b) - 4 h(1+b) = b^2 + 4b + 3 - 4 h(1+b) = b^2 + 4b - 1 Next, find h(1): h(1) = (1)^2 + 2(1) - 4 h(1) = 1 + 2 - 4 h(1) = -1 Now, substitute these into the expression: (h(1+b)-h(1))/(b) = ((b^2 + 4b - 1) - (-1))/(b) (h(1+b)-h(1))/(b) = (b^2 + 4b - 1 + 1)/(b) (h(1+b)-h(1))/(b) = (b^2 + 4b)/(b) Factor out b from the numerator: (h(1+b)-h(1))/(b) = (b(b+4))/(b) Cancel b (assuming b ≠ 0): (h(1+b)-h(1))/(b) = b+4 The simplified expression is b+4. Step 5: Solve part d). d) Find (h(a+b)-h(a))/(b). First, find h(a+b): h(a+b) = (a+b)^2 + 2(a+b) - 4 h(a+b) = a^2 + 2ab + b^2 + 2a + 2b - 4 Next, find h(a): h(a) = a^2 + 2a - 4 Now, substitute these into the expression: (h(a+b)-h(a))/(b) = ((a^2 + 2ab + b^2 + 2a + 2b - 4) - (a^2 + 2a - 4))/(b) Distribute the negative sign in the numerator: (h(a+b)-h(a))/(b) = (a^2 + 2ab + b^2 + 2a + 2b - 4 - a^2 - 2a + 4)/(b) Combine like terms in the numerator: (h(a+b)-h(a))/(b) = ((a^2 - a^2) + (2a - 2a) + (-4 + 4) + 2ab + b^2 + 2b)/(b) (h(a+b)-h(a))/(b) = (2ab + b^2 + 2b)/(b) Factor out b from the numerator: (h(a+b)-h(a))/(b) = (b(2a + b + 2))/(b) Cancel b (assuming b ≠ 0): (h(a+b)-h(a))/(b) = 2a + b + 2 The simplified expression is 2a + b + 2.