Question

the image shows a mathematical expression, but due to rotation and clarity issues, the exact expression is hard to parse. visible elements include parentheses, an equal sign, and some characters that might be part of a variable (like x or similar) and exponents or coefficients, but the full problem is not clear from the rotated and partially visible text.

Explanation:

Step1: Analyze the equation structure

The equation seems to be \( 4(H)=5H^{3}-H \) (assuming the characters are interpreted as such, maybe a typo or formatting issue, but we'll work with this structure). First, we want to bring all terms to one side to form a polynomial equation.

Step2: Rearrange the equation

Subtract \( 4H \) from both sides: \( 0 = 5H^{3}-H - 4H \), which simplifies to \( 5H^{3}-5H = 0 \).

Step3: Factor the polynomial

Factor out \( 5H \): \( 5H(H^{2}-1)=0 \). Then, using the difference of squares \( a^{2}-b^{2}=(a - b)(a + b) \), we get \( 5H(H - 1)(H + 1)=0 \).

Step4: Solve for H

Set each factor equal to zero:

• \( 5H = 0 \) gives \( H = 0 \)
• \( H - 1 = 0 \) gives \( H = 1 \)
• \( H + 1 = 0 \) gives \( H = - 1 \)

Answer:

The solutions for \( H \) are \( H = -1 \), \( H = 0 \), and \( H = 1 \)