7. Simplify, State any restrictions on the variables. (3)/(x+1)+(1)/(x^2)-3x-4 b) (2t)/(t-4)-(5t)/(t^2)-16 c) (3)/(t^2)+t-6+(5)/((t+3)^2) d) (4x)/(x^2)+6x+8-(3x)/(x^2)-3x-10 c) (x-1)/(x^2)-9+(x+7)/(x^2)-5x+6 (2t+1)/(2t^2)-14t+24+(5t)/(4t^2)-8t-12

## Step 1: Factor the denominators<br />### Factor each denominator to find the least common denominator (LCD). For example, $x^2 - 3x - 4 = (x-4)(x+1)$ and $t^2 - 16 = (t-4)(t+4)$. Similarly, $t^2 + t - 6 = (t+3)(t-2)$, $x^2 + 6x + 8 = (x+4)(x+2)$, $x^2 - 3x - 10 = (x-5)(x+2)$, $x^2 - 9 = (x-3)(x+3)$, $x^2 - 5x + 6 = (x-2)(x-3)$, $2t^2 - 14t + 24 = 2(t-3)(t-4)$, and $4t^2 - 8t - 12 = 4(t-3)(t+1)$.<br /><br />## Step 2: Find the least common denominator (LCD)<br />### Identify the LCD by considering all unique factors in the denominators. For a), the LCD is $(x+1)(x-4)$. For b), the LCD is $(t-4)(t+4)$. For c), the LCD is $(t+3)^2(t-2)$. For d), the LCD is $(x+4)(x+2)(x-5)$. For e), the LCD is $(x-3)(x+3)(x-2)$. For f), the LCD is $4(t-3)(t-4)(t+1)$.<br /><br />## Step 3: Rewrite each fraction with the LCD<br />### Multiply each fraction by a form of 1 so that each fraction has the LCD. For example, in a), multiply $\frac{3}{x+1}$ by $\frac{x-4}{x-4}$.<br /><br />## Step 4: Combine the numerators<br />### Add or subtract the numerators over the common denominator. Simplify the resulting numerator by combining like terms.<br /><br />## Step 5: Simplify the resulting fraction<br />### Factor the numerator if possible and cancel any common factors between the numerator and denominator.<br /><br />## Step 6: State any restrictions<br />### Identify any values of the variable that would make the denominator zero in the original expression or any intermediate step. These values are the restrictions on the variable.