`(2d^2-d-10)/(d^2+7d+10)=(d^2-4d+3)/(d^2+2d-15)`
In the equation above, what is the value of d?

Options

A. -4
B. 2
C. 4
D. 6

Answer

To solve the equation

\[ \frac{2d^2 - d - 10}{d^2 + 7d + 10} = \frac{d^2 - 4d + 3}{d^2 + 2d - 15}, \]

we will factor the quadratic expressions in both the numerator and the denominator.

### Step 1: Factor the left-hand side numerator \(2d^2 - d - 10\)

We need to find two numbers that multiply to \(2 \times (-10) = -20\) and add to \(-1\). The numbers \(-5\) and \(4\) satisfy this condition.

So, we can rewrite:

\[ 2d^2 - d - 10 = 2d^2 - 5d + 4d - 10 = (2d^2 - 5d) + (4d - 10). \]

Factoring by grouping:

\[ = d(2d - 5) + 2(2d - 5) = (2d - 5)(d + 2). \]

### Step 2: Factor the left-hand side denominator \(d^2 + 7d + 10\)

We need two numbers that multiply to \(10\) and add to \(7\). The numbers \(5\) and \(2\) work.

So, we can write:

\[ d^2 + 7d + 10 = (d + 5)(d + 2). \]

### Step 3: Factor the right-hand side numerator \(d^2 - 4d + 3\)

We need two numbers that multiply to \(3\) and add to \(-4\). The numbers \(-3\) and \(-1\) work.

So, we can write:

\[ d^2 - 4d + 3 = (d - 3)(d - 1). \]

### Step 4: Factor the right-hand side denominator \(d^2 + 2d - 15\)

We need two numbers that multiply to \(-15\) and add to \(2\). The numbers \(5\) and \(-3\) work.

So, we can write:

\[ d^2 + 2d - 15 = (d + 5)(d - 3). \]

### Step 5: Rewrite the equation with factored forms

Now we can rewrite the original equation:

\[ \frac{(2d - 5)(d + 2)}{(d + 5)(d + 2)} = \frac{(d - 3)(d - 1)}{(d + 5)(d - 3)}. \]

### Step 6: Cancel common factors

We can cancel \((d + 2)\) from the left-hand side and \((d - 3)\) from the right-hand side (assuming \(d \neq -2\) and \(d \neq 3\)):

\[ \frac{2d - 5}{d + 5} = \frac{d - 1}{d + 5}. \]

### Step 7: Cross-multiply

Cross-multiplying gives us:

\[ (2d - 5)(d + 5) = (d - 1)(d + 5). \]

### Step 8: Expand both sides

Expanding both sides:

Left-hand side:

\[ 2d^2 + 10d - 5d - 25 = 2d^2 + 5d - 25. \]

Right-hand side:

\[ d^2 + 5d - d - 5 = d^2 + 4d - 5. \]

### Step 9: Set the equation to zero

Now we have:

\[ 2d^2 + 5d - 25 = d^2 + 4d - 5. \]

Rearranging gives:

\[ 2d^2 - d^2 + 5d - 4d - 25 + 5 = 0, \]

which simplifies to:

\[ d^2 + d - 20 = 0. \]

### Step 10: Factor the resulting quadratic

We need two numbers that multiply to \(-20\) and add to \(1\). The numbers \(5\) and \(-4\) work.

So, we can write:

\[ d^2 + 5d - 4d - 20 = (d + 5)(d - 4) = 0. \]

### Step 11: Solve for \(d\)

Setting each factor to zero gives:

\[ d + 5 = 0 \quad \Rightarrow \quad d = -5, \]
\[ d - 4 = 0 \quad \Rightarrow \quad d = 4. \]

### Final Solution

The possible values of \(d\) are \(-5\) and \(4\). Since we need to check for any restrictions from the original equation, we find that \(d = -5\) would make the denominator zero, so we discard that solution.

Thus, the final answer is:

\[ \boxed{4}. \]