5) $$ 10 m^{2}-13 m n-3 n^{2} $$

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UpStudy AI · Accepted Answer

To factor the quadratic expression $$ 10m^{2} - 13mn - 3n^{2} $$, we'll look for two binomials of the form $$(am + bn)(cm + dn)$$ that multiply to give the original expression.

### Step 1: Identify the Coefficients
We need to find $$a$$, $$b$$, $$c$$, and $$d$$ such that:

$$
\begin{align*}
ac &= 10 \quad (\text{coefficient of } m^2) \
bd &= -3 \quad (\text{constant term}) \
ad + bc &= -13 \quad (\text{coefficient of } mn)
\end{align*}
$$

### Step 2: Determine Possible Values
Let's consider the possible pairs for $$a$$ and $$c$$ that satisfy $$ac = 10$$:

- $$a = 5$$, $$c = 2$$
- $$a = 10$$, $$c = 1$$

Similarly, for $$b$$ and $$d$$ with $$bd = -3$$:

- $$b = 1$$, $$d = -3$$
- $$b = -1$$, $$d = 3$$
- $$b = 3$$, $$d = -1$$
- $$b = -3$$, $$d = 1$$

### Step 3: Find the Correct Combination
Let's try $$a = 5$$ and $$c = 2$$:

1. **Option 1: $$b = 1$$, $$d = -3$$**
   $$
   ad + bc = (5)(-3) + (1)(2) = -15 + 2 = -13
   $$
   This satisfies the condition $$ad + bc = -13$$.

2. **Other options** do not satisfy the condition.

### Step 4: Write the Factored Form
Using the values $$a = 5$$, $$b = 1$$, $$c = 2$$, and $$d = -3$$, the factored form is:

$$
(5m + n)(2m - 3n)
$$

### Verification
Multiply the binomials to ensure correctness:
$$
\begin{align*}
(5m + n)(2m - 3n) &= 5m \cdot 2m + 5m \cdot (-3n) + n \cdot 2m + n \cdot (-3n) \
&= 10m^2 - 15mn + 2mn - 3n^2 \
&= 10m^2 - 13mn - 3n^2
\end{align*}
$$
This matches the original expression.

**Final Answer:**
$$
10m^{2} - 13mn - 3n^{2} = (5m + n)(2m - 3n)
$$
$$10m^{2} - 13mn - 3n^{2} = (5m + n)(2m - 3n)$$

5) \( 10 m^{2}-13 m n-3 n^{2} \)

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