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Answer
The factored form is \((x+5)(5x-2)\).
Solution
Factor the expression \( 5x^2+23x-10 \).
Factor the expression by following steps:
- step0: Factor:
\(5x^{2}+23x-10\)
- step1: Rewrite the expression:
\(5x^{2}+\left(-2+25\right)x-10\)
- step2: Calculate:
\(5x^{2}-2x+25x-10\)
- step3: Rewrite the expression:
\(x\times 5x-x\times 2+5\times 5x-5\times 2\)
- step4: Factor the expression:
\(x\left(5x-2\right)+5\left(5x-2\right)\)
- step5: Factor the expression:
\(\left(x+5\right)\left(5x-2\right)\)
The factored form of the expression \(5x^2+23x-10\) is \((x+5)(5x-2)\).
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Simplify this solution Beyond the Answer
To factor the quadratic expression \( 5 x^2 + 23 x - 10 \), we look for two numbers that multiply to \( 5 \times (-10) = -50 \) and add up to \( 23 \). After some trial and error, we find that \( 25 \) and \( -2 \) fit the bill. Therefore, we rewrite the expression: \[ 5 x^2 + 25 x - 2 x - 10 \] Now, we factor by grouping: \[ (5 x^2 + 25 x) + (-2 x - 10) \] This gives us: \[ 5x(x + 5) - 2(x + 5) \] Factoring out the common factor \( (x + 5) \): \[ (5x - 2)(x + 5) \] Thus, the completely factored form is: \[ \boxed{(5x - 2)(x + 5)} \]
